" .>Daoversion2"^{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 16 3 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Fix ed Width" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 17 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 51 " \+ Introduction to the VecCalc Package -- Version 8.0" }}{PARA 0 "" 0 "" {TEXT -1 49 "The possible HELP pages are listed at the bottom." }} {PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 17 " command(args) " }}{PARA 257 "" 0 "" {TEXT -1 25 " VecCalc[command](args)" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 7 "VecCalc" }{TEXT -1 113 " package is a collec tion of commands designed to simplify calculations which arise from ve ctor calculus problems." }}{PARA 15 "" 0 "" {TEXT -1 41 "To load the p ackage, execute the command " }{TEXT 261 14 "with(VecCalc);" }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 7 "VecCalc" }{TEXT -1 42 " pac kage automatically loads the packages:" }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "student" 2 "student" "" }{TEXT -1 3 " " } {HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 12 "Many of the " }{TEXT 262 7 "VecCalc" }{TEXT -1 74 " comma nds have shorter aliases which become available after executing the " }{TEXT 263 7 "VecCalc" }{TEXT -1 9 " command " }{HYPERLNK 17 "VCalias " 2 "VCalias" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 264 7 "VecCalc" }{TEXT -1 194 " commands are divided into sever al groups. These commands are listed below by group and are followed \+ by the alias in parentheses, if there is one. Each command is a hyper link to its help page." }}{PARA 16 "" 0 "" {TEXT -1 21 "Environment se ttings:" }}{PARA 258 "" 0 "" {HYPERLNK 17 "VCalias" 2 "VCalias" "" } {TEXT -1 6 " " }{HYPERLNK 17 "OutputVectorType" 2 "OutputTypes" " " }{TEXT -1 3 " " }{HYPERLNK 17 "OutputMatrixType" 2 "OutputTypes" " " }{TEXT -1 1 " " }}{PARA 16 "" 0 "" {TEXT -1 58 "Commands to make and evaluate scalar and vector functions:" }}{PARA 258 "" 0 "" {HYPERLNK 17 "MakeFunction" 2 "MakeFunction" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "&->" 2 "MakeFunction" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "MF" 2 " MakeFunction" "" }{TEXT -1 9 " ) " }{HYPERLNK 17 "evalFunction" 2 "evalFunction" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "&@" 2 "evalFunct ion" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "EF" 2 "evalFunction" "" } {TEXT -1 2 " )" }}{PARA 16 "" 0 "" {TEXT -1 48 "Commands to perform li near algebra computations:" }}{PARA 258 "" 0 "" {HYPERLNK 17 "Dot" 2 " Dot" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "&." 2 "dot" "" }{TEXT -1 6 " " }{HYPERLNK 17 "Length" 2 "len" "" }{TEXT -1 6 " " } {HYPERLNK 17 "Angle" 2 "Angle" "" }{TEXT -1 6 " " }{HYPERLNK 17 " Cross" 2 "Cross" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "&x" 2 "cross" " " }{TEXT -1 6 " " }{HYPERLNK 17 "simplifyvec" 2 "simplifyvec" "" }}{PARA 258 "" 0 "" {HYPERLNK 17 "Determinant" 2 "Determinant" "" } {TEXT -1 4 " ( " }{HYPERLNK 17 "Det" 2 "Determinant" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "LeadingPrincipalMinorDeterminants" 2 "LeadingPri ncipalMinorDeterminants" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "LPMD" 2 "LeadingPrincipalMinorDeterminants" "" }{TEXT -1 2 " )" }}{PARA 16 "" 0 "" {TEXT -1 59 "Commands to change angular measure and coordinates: \+ (See " }{HYPERLNK 17 "AngleConversion" 2 "AngleConversion" "" } {TEXT -1 3 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2 D" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConv ersion3D" "" }{TEXT -1 2 ".)" }}{PARA 258 "" 0 "" {HYPERLNK 17 "deg2ra d" 2 "AngleConversions" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "d2r" 2 "A ngleConversions" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "rad2deg" 2 " AngleConversions" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2d" 2 "AngleCo nversions" "" }{TEXT -1 5 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "po lar2rect" 2 "CoordConversion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "p 2r" 2 "CoordConversion2D" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "rect2p olar" 2 "CoordConversion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2p" 2 "CoordConversion2D" "" }{TEXT -1 2 " )" }}{PARA 258 "" 0 "" {HYPERLNK 17 "cyl2rect" 2 "CoordConversion3D" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "c2r" 2 "CoordConversion3D" "" }{TEXT -1 7 " ) " } {HYPERLNK 17 "rect2cyl" 2 "CoordConversion3D" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "r2c" 2 "CoordConversion3D" "" }{TEXT -1 4 " ) " }} {PARA 258 "" 0 "" {HYPERLNK 17 "sph2rect" 2 "CoordConversion3D" "" } {TEXT -1 4 " ( " }{HYPERLNK 17 "s2r" 2 "CoordConversion3D" "" }{TEXT -1 7 " ) " }{HYPERLNK 17 "rect2sph" 2 "CoordConversion3D" "" } {TEXT -1 4 " ( " }{HYPERLNK 17 "r2s" 2 "CoordConversion3D" "" }{TEXT -1 4 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "sph2cyl" 2 "CoordConvers ion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "s2c " 2 "CoordConversion3D " "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "cyl2sph" 2 "CoordConversion 3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "c2s" 2 "CoordConversion3D" " " }{TEXT -1 5 " ) " }}{PARA 16 "" 0 "" {TEXT -1 36 "Commands to anal yse a curve: (See " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 " .)" }}{PARA 258 "" 0 "" {HYPERLNK 17 "CurveVelocity" 2 "Curve" "" } {TEXT -1 4 " ( " }{HYPERLNK 17 "Cv" 2 "Curve" "" }{TEXT -1 6 " ) \+ " }{HYPERLNK 17 "CurveAcceleration" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Ca" 2 "Curve" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "Curv eJerk" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Cj" 2 "Curve" " " }{TEXT -1 6 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "CurveTangent " 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CT" 2 "Curve" "" } {TEXT -1 7 " ) " }{HYPERLNK 17 "CurveNormal" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CN" 2 "Curve" "" }{TEXT -1 11 " ) \+ " }{HYPERLNK 17 "CurveBinormal" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "CB" 2 "Curve" "" }{TEXT -1 2 " )" }}{PARA 258 "" 0 "" {HYPERLNK 17 "CurveSpeed" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Cs" 2 "Curve" "" }{TEXT -1 9 " ) " }{HYPERLNK 17 "CurveArcLe ngth" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CL" 2 "Curve" "" }{TEXT -1 29 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "CurveCurvature" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Ck" 2 "Curve" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "Curv eTorsion" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Ct" 2 "Curve " "" }{TEXT -1 31 " ) " }}{PARA 258 "" 0 " " {HYPERLNK 17 "CurveTangentialAcceleration" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CaT" 2 "Curve" "" }{TEXT -1 8 " ) " } {HYPERLNK 17 "CurveNormalAcceleration" 2 "Curve" "" }{TEXT -1 4 " ( \+ " }{HYPERLNK 17 "CaN" 2 "Curve" "" }{TEXT -1 2 " )" }}{PARA 258 "" 0 " " {HYPERLNK 17 "CurveForget" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Cforget" 2 "Curve" "" }{TEXT -1 52 " ) \+ " }}{PARA 16 "" 0 "" {TEXT -1 38 "Comm ands to analyse a surface: (See " }{HYPERLNK 17 "Surface" 2 "Surface " "" }{TEXT -1 2 ".)" }}{PARA 258 "" 0 "" {HYPERLNK 17 "SurfaceTangent s" 2 "Surface" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "ST" 2 "Surface" " " }{TEXT -1 10 " ) " }{HYPERLNK 17 "SurfaceNormal" 2 "Surface" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "SN" 2 "Surface" "" }{TEXT -1 2 " \+ )" }}{PARA 258 "" 0 "" {HYPERLNK 17 "SurfaceNormalLength" 2 "Surface" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "SNL" 2 "Surface" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "SurfaceArea" 2 "Surface" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "SA" 2 "Surface" "" }{TEXT -1 4 " ) " }}{PARA 258 "" 0 " " {HYPERLNK 17 "SurfaceForget" 2 "Surface" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Sforget" 2 "Surface" "" }{TEXT -1 28 " ) \+ " }}{PARA 16 "" 0 "" {TEXT -1 45 "Commands for differential \+ operations: (See " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Mu ltiMaxMin" 2 "MultiMaxMin" "" }{TEXT -1 2 ".) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Grad" 2 "Gradient" "" }{TEXT -1 12 " ) \+ " }{HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Hess" 2 "Hessian" "" }{TEXT -1 14 " ) " }} {PARA 258 "" 0 "" {HYPERLNK 17 "Divergence" 2 "Divergence" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Div" 2 "Divergence" "" }{TEXT -1 11 " ) \+ " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 6 " " } {HYPERLNK 17 "Laplacian" 2 "Laplacian" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Lap" 2 "Laplacian" "" }{TEXT -1 3 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "JacobianMatrix" 2 "JacobianMatrix" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Jac" 2 "JacobianMatrix" "" }{TEXT -1 7 " ) " }{HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "JDet" 2 "JacobianDeterminant" "" }{TEXT -1 2 " )" }}{PARA 258 "" 0 "" {HYPERLNK 17 "ScalarPotential" 2 "ScalarPot en tial" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "SPot" 2 "ScalarPotential " "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "VectorPotential" 2 "VectorPote ntial" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "VPot" 2 "VectorPotential" "" }{TEXT -1 6 " ) " }}{PARA 16 "" 0 "" {TEXT -1 33 "Commands for i ntegral operations:" }}{PARA 258 "" 0 "" {HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Muint" 2 "Multiplei nt" "" }{TEXT -1 13 " ) " }{HYPERLNK 17 "multipleint" 2 "Mul tipleint" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "muint" 2 "Multipleint" "" }{TEXT -1 5 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "LineIntScalar Inert" 2 "LineIntScalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Lis" 2 " LineIntScalar" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "LineIntScalar " 2 "LineIntScalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "lis" 2 "LineI ntScalar" "" }{TEXT -1 5 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "Lin eIntVectorInert" 2 "LineIntVector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Liv" 2 "LineIntVector" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "Li neIntVector" 2 "LineIntVector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "li v" 2 "LineIntVector" "" }{TEXT -1 5 " ) " }}{PARA 258 "" 0 "" {HYPERLNK 17 "SurfaceIntScalarInert" 2 "SurfaceIntScalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Sis" 2 "SurfaceIntScalar" "" }{TEXT -1 5 " \+ ) " }{HYPERLNK 17 "SurfaceIntScalar" 2 "SurfaceIntScalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "sis" 2 "SurfaceIntScalar" "" }{TEXT -1 2 " \+ )" }}{PARA 258 "" 0 "" {HYPERLNK 17 "SurfaceIntVectorInert" 2 "Surface IntVector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Siv" 2 "SurfaceIntVect or" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "SurfaceIntVector" 2 "Surface IntVector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "siv" 2 "SurfaceIntVect or" "" }{TEXT -1 2 " )" }}{PARA 15 "" 0 "" {TEXT -1 43 "The following \+ commands are internal to the " }{TEXT 271 7 "VecCalc" }{TEXT -1 41 " p ackage and are unavailable to the user:" }}{PARA 258 "" 0 "" {TEXT -1 59 "getvectype mapfunc map_unapply Get_Vars MuInt_noChk" }} {PARA 15 "" 0 "" {TEXT -1 93 "The following commands were modified to \+ map onto lists, Vectors, Matrices and Arrays: (See " }{HYPERLNK 17 " MappedFunctions" 2 "MappedFunctions" "" }{TEXT -1 2 ".)" }}{PARA 258 " " 0 "" {HYPERLNK 17 "Limit" 2 "MappedFunctions" "" }{TEXT -1 3 " " } {HYPERLNK 17 "limit" 2 "MappedFunctions" "" }{TEXT -1 6 " " } {HYPERLNK 17 "Diff" 2 "MappedFunctions" "" }{TEXT -1 3 " " } {HYPERLNK 17 "diff" 2 "MappedFunctions" "" }{TEXT -1 3 " " } {HYPERLNK 17 "D" 2 "MappedFunctions" "" }{TEXT -1 6 " " } {HYPERLNK 17 "Int" 2 "MappedFunctions" "" }{TEXT -1 3 " " } {HYPERLNK 17 "int" 2 "MappedFunctions" "" }{TEXT -1 6 " " } {HYPERLNK 17 "value" 2 "MappedFunctions" "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "To calculate the dot product of two vectors A and B, use " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "wi th(VecCalc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A:=; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")s#4H\"-%'MATR IXG6#7%7#%\"xG7#%\"yG7#%\"zG&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "B:=<1,2,3>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")_$4H\"-%'MATRIXG6#7%7#\"\"\"7#\"\"#7#\"\"$&%'V ectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Dot(A,B );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"*&\"\"#F%%\"yGF%F%* &\"\"$F%%\"zGF%F%" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 17 "Acknowledg ements:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 215 "The commands in this package are used extensively throughout the text \"Multivaria ble CalcLabs with Maple\" by Arthur Belmonte and Philip B. Yasskin, pu blished by Brooks/Cole in several editions with different titles:" }} {PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 265 7 "VecCalc" }{TEXT -1 784 " commands were originally written for Maple V Release 3 by A. Bel monte and P. Yasskin. The commands were organized into a package for \+ Maple V Release 3 by James Warren and P. Yasskin. The help pages were first written for Maple V Release 3 by David Arnold, J. Warren and P. Yasskin and converted to Maple V Release 4 by Ken Parker, Jared Teslo w and P. Yasskin. The package was converted to a module and the comma nds were updated to Maple 6 and 7 by A. Belmonte and the help pages we re updated to Maple 6 and 7 by P. Yasskin. The module, commands and h elp pages were significantly modified for Maple 8 to be compatible wit h new Vector and Matrix types by P. Yasskin with help from Chad Wellin gton, Krista Rister, Chris Haag, Ethan McConnell, Allison DenBleyker a nd Jeffrey Yasskin." }}{PARA 15 "" 0 "" {TEXT -1 137 "@ Copyright 1995 -2003 by Arthur Belmonte and Philip B. Yasskin, Department of Mathemat ics, Texas A&M University with all rights reserved." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "libname" 2 "libname " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "with" 2 "with" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "MappedFunctions" 2 "MappedFunctions" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface" 2 "Surface" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "D iffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin" 2 "MultiMaxMin " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConver sion2D" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "Coor dConversion3D" "" }{TEXT -1 44 ". To get help on any command, you can \+ type ?" }{TEXT 258 7 "command" }{TEXT -1 5 " or ?" }{TEXT 266 7 "VecCa lc" }{TEXT -1 1 "[" }{TEXT 256 7 "command" }{TEXT -1 9 "] (where " } {TEXT 257 7 "command" }{TEXT -1 64 " is from the above list). See the Help Cross-Referencing below." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Help Cross-Referencing:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 151 "Here is the list of all the help pages and the list of help to pics which should point to each of them. Just click on one of the mai n help pages below." }}{PARA 0 "" 0 "" {TEXT -1 70 "__________________ ____________________________________________________" }}{PARA 15 "" 0 "" {TEXT -1 20 "VecCalc -- this page" }}{PARA 15 "" 0 "" {HYPERLNK 17 "VCalias" 2 "VCalias" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "OutputTypes " 2 "OutputTypes" "" }}{PARA 0 "" 0 "" {TEXT 267 19 " OutputVectorTy pe" }}{PARA 0 "" 0 "" {TEXT 268 19 " OutputMatrixType" }}{PARA 15 " " 0 "" {HYPERLNK 17 "MakeFunction" 2 "MakeFunction" "" }}{PARA 256 "" 0 "" {TEXT -1 14 " &-> MF" }}{PARA 15 "" 0 "" {HYPERLNK 17 "eva lFunction" 2 "evalFunction" "" }}{PARA 256 "" 0 "" {TEXT -1 14 " &@ \+ EF" }}{PARA 15 "" 0 "" {HYPERLNK 17 "MappedFunctions" 2 "MappedF unctions" "" }}{PARA 256 "" 0 "" {TEXT -1 17 " Limit limit" }} {PARA 256 "" 0 "" {TEXT -1 20 " Diff diff D" }}{PARA 256 "" 0 "" {TEXT -1 15 " Int int" }}{PARA 256 "" 0 "" {TEXT -1 8 " va lue" }}{PARA 256 "" 0 "" {TEXT 272 70 "_______________________________ _______________________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Dot" 2 "Dot" "" }}{PARA 256 "" 0 "" {TEXT -1 5 " &." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Length" 2 "len" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Angle" 2 "Angle" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Cross" 2 "Cro ss" "" }{HYPERLNK 17 "" 2 "cross" "" }}{PARA 256 "" 0 "" {TEXT -1 5 " \+ &x" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Determinant" 2 "LinearAlgebra:- Determinant" "" }}{PARA 0 "" 0 "" {TEXT 269 6 " Det" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LeadingPrincipalMinorDeterminants" 2 "LeadingPrinci palMinorDeterminants" "" }}{PARA 0 "" 0 "" {TEXT 270 7 " LPMD" }} {PARA 15 "" 0 "" {HYPERLNK 17 "simplifyvec" 2 "simplifyvec" "" }} {PARA 0 "" 0 "" {TEXT -1 70 "_________________________________________ _____________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "AngleC onversions" 2 "AngleConversions" "" }{TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 19 " deg2rad d2r" }}{PARA 256 "" 0 "" {TEXT -1 19 " \+ rad2deg r2d" }}{PARA 15 "" 0 "" {HYPERLNK 17 "CoordConversion2D " 2 "CoordConversion2D" "" }}{PARA 256 "" 0 "" {TEXT -1 19 " polar2r ect p2r" }}{PARA 256 "" 0 "" {TEXT -1 19 " rect2polar r2p" }} {PARA 15 "" 0 "" {HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D " "" }}{PARA 256 "" 0 "" {TEXT -1 19 " cyl2rect c2r" }}{PARA 256 "" 0 "" {TEXT -1 19 " rect2cyl r2c" }}{PARA 256 "" 0 "" {TEXT -1 19 " sph2rect s2r" }}{PARA 256 "" 0 "" {TEXT -1 19 " \+ rect2sph r2s" }}{PARA 256 "" 0 "" {TEXT -1 19 " sph2cyl s2c " }}{PARA 256 "" 0 "" {TEXT -1 19 " cyl2sph c2s" }}{PARA 0 "" 0 "" {TEXT -1 70 "____________________________________________________ __________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Curve" 2 "Curve" "" }}{PARA 256 "" 0 "" {TEXT -1 9 " frenet" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveVelocity Cv" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveAcceleration Ca" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveJerk Cj" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveTangent CT" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveNormal  CN" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveBinormal CB" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveSpeed Cs" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveArcLength CL" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveCurvature Ck" }}{PARA 256 "" 0 "" {TEXT -1 34 " CurveTorsion Ct" }}{PARA 256 "" 0 "" {TEXT -1 35 " CurveTangentialAcceleration CaT" }}{PARA 256 "" 0 "" {TEXT -1 35 " CurveNormalAcceleration CaN" }}{PARA 15 "" 0 "" {HYPERLNK 17 "CurveForget" 2 "CurveForget" "" }}{PARA 256 "" 0 "" {TEXT -1 10 " Cforget" }}{PARA 0 "" 0 "" {TEXT -1 70 "______________ ________________________________________________________" }}{PARA 15 " " 0 "" {HYPERLNK 17 "Surface" 2 "Surface" "" }}{PARA 256 "" 0 "" {TEXT -1 26 " SurfaceTangents ST" }}{PARA 256 "" 0 "" {TEXT -1 26 " SurfaceNormal SN" }}{PARA 256 "" 0 "" {TEXT -1 27 " Su rfaceNormalLength SNL" }}{PARA 256 "" 0 "" {TEXT -1 26 " SurfaceAre a  SA" }}{PARA 15 "" 0 "" {HYPERLNK 17 "SurfaceForget" 2 "Surf aceForget" "" }}{PARA 256 "" 0 "" {TEXT -1 10 " Sforget" }}{PARA 0 " " 0 "" {TEXT -1 70 "__________________________________________________ ____________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Diffops" 2 "Dif fops" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "MultiMaxMin" 2 "MultiMaxMin " "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Gradient" 2 "Gradient" "" } {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 7 " Grad" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Hessian" 2 "Hessian" "" }}{PARA 256 "" 0 "" {TEXT -1 7 " Hess" }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Diverge nce" 2 "Divergence" "" }}{PARA 256 "" 0 "" {TEXT -1 6 " Div" }} {PARA 15 "" 0 "" {HYPERLNK 17 "Curl" 2 "Curl" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Laplacian" 2 "Laplacian" "" }}{PARA 256 "" 0 "" {TEXT -1 6 " Lap" }}{PARA 15 "" 0 "" {HYPERLNK 17 "JacobianMatrix" 2 "Jaco bianMatrix" "" }}{PARA 256 "" 0 "" {TEXT -1 6 " Jac" }}{PARA 15 "" 0 "" {HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }} {PARA 256 "" 0 "" {TEXT -1 7 " JDet" }}{PARA 15 "" 0 "" {HYPERLNK 17 "ScalarPotential" 2 "ScalarPotential" "" }}{PARA 256 "" 0 "" {TEXT -1 7 " SPot" }}{PARA 15 "" 0 "" {HYPERLNK 17 "VectorPotential" 2 "Ve ctorPotential" "" }}{PARA 257 "" 0 "" {TEXT -1 7 " VPot" }}{PARA 0 " " 0 "" {TEXT -1 70 "__________________________________________________ ____________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }}{PARA 256 "" 0 "" {TEXT -1 22 " Mui nt" }}{PARA 256 "" 0 "" {TEXT -1 22 " multipleint muint" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LineIntScalar" 2 "LineIntScalar" "" }}{PARA 256 "" 0 "" {TEXT -1 29 " LineIntScalarInert Lis" }}{PARA 256 " " 0 "" {TEXT -1 29 " lis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LineIntVector" 2 "LineIntVector" "" }}{PARA 256 "" 0 "" {TEXT -1 29 " LineIntVectorInert Liv" }}{PARA 256 "" 0 "" {TEXT -1 29 " liv" }}{PARA 15 "" 0 "" {HYPERLNK 17 "SurfaceIntScalar" 2 "SurfaceIntScalar" "" }}{PARA 256 "" 0 "" {TEXT -1 29 " SurfaceIntScalarInert Sis" }}{PARA 256 "" 0 "" {TEXT -1 29 " sis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surfa ceIntVector" 2 "SurfaceIntVector" "" }}{PARA 256 "" 0 "" {TEXT -1 29 " SurfaceIntVectorInert Siv" }}{PARA 256 "" 0 "" {TEXT -1 29 " \+ siv" }}{PARA 0 "" 0 "" {TEXT -1 70 "______________ ________________________________________________________" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12909272 12909352 }{RTABLE M7R0 I5RTABLE_SAVE/12909272X*%)anythingG6"6"[gl!#%!!!"$"$%"xG%"yG%"zG6" } {RTABLE M7R0 I5RTABLE_SAVE/12909352X*%)anythingG6"6"[gl!#%!!!"$"$"""""#""$6" } lis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LineIntVector" 2 "LineIntVector" "" }}{PARA 256 "" 0 "" {TEXT -1 29 " LineIntVectorInert Liv" }}{PARA 256 "" 0 "" {TEXT -1 29 " liv" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surfac" VecCalc" VCalias" OutputTypes" simplifyvec"MappedFunctions"MakeFunction"evalFunction"Dot"Length"Angle"Cross" Determinant"#LeadingPrincipalMinorDeterminants"AngleConversion"CoordConversion2D"CoordConversion3D"Curve" CurveForget" Surface"SurfaceForget" Diffops" MultiMaxMin" Gradient" Hessian" Divergence"Curl" Laplacian"JacobianMatrix"JacobianDeterminant"ScalarPotential"VectorPotential" Multipleint"LineIntScalar"LineIntVector"SurfaceIntScalar"SurfaceIntVector"index,package"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"VecCalc"Curve"VecCalc"Surface"VecCalc"VecCalc"Diffops"Diffops"Diffops"Diffops"Diffops"Diffops"Diffops"Diffops"Diffops"VecCalc"VecCalc"VecCalc"VecCalc"VecCalcdeterm ""determi"determin# """""""" determinan "" determinant#'""" " """"detg"df ""dfr"dfrgf"dg"""""dgf ""di"""dif"diff """differ/""""""""""" differential"""" differentiat ""diffop7"""""""""""""dimen"dimens""" dimensional"""""dinat"disp"displa """""display""""""distanc """ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalwarnoutputbulletitemcommandveccalcevalfunctevaluatfunctionusingpackagoperatorevaluatefunctaliacanusedafterexecutvcaliaefcallsequencfncpointcautmayneedspacelseaplethinksubsequsymbolpartnameexpresslistarrayouencloparenthesparameterformproducmakefunctlistvectorrepresentatfunctidescriptionunnecessarlistlistvalufunctionhowevessentialectormatrixtheseonlyperformwithveccalcalwayaccesslongormexamplvariablcurvrgrtablegmatrixgtgoperatorgarrowgfvectorgcolumngqmcolumngseveralvariablfieldcoordinattransformatparamtricsurfacmffgseugvggfalongagxgygzgfafakfnfnglinelinegsgskcolumngfbbsgfcolumncopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopneutralprecedencrtablsaveanythingggl"3ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalcsimplifyvecsimpliflislistlistvectormatrixperformalgebracallsequencexprparameteralgebraicexpressinvolvlistlistlistmatricdescriptsimplifyvecsimplificattshispartpackagcanusedformonlyaftercommandwithalwayaccesslongexamplugvgxgygzgsqrtxgfygfzgfagrtablegcrmatrixgbgcgdgmatrixgfagfbgfcgfdgfcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoconvertrtablsaveanythinggglaftalia@ andeterminantrararthurbecomblishca.ccosg/clickYcommandconvert|cosTcros#curvetangentialacceleratodegrepdetdistanc0ed?equalS evalfunctexpgsfgfaforgetbfunctionAgradiheadhilipibmjinertsurfaceintvectorR intvector| jacobianmat}lapilesslineintRlivmake}mathqmayOmuintgPndsnordeterminantntolaropeoriginoutputpackagQpartphipreviouPrectresultrierl rpotentialsagsforgetksivgQspotsum surfacenormalteslotheta titl ttingupdatcvaveccad vectorpotwityasskin"Cibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalcdeterminantcalculatmatrixaliacanusedafterexecutvcaliacommanddetallingsequencveccalcparametersquarlistdescriptdeterminanosexactsamelinearalgebraexceptalsoactwellpartpackagformdeterminantonlyperformwithionalwayaccesslongexamplvandermondematrixmgrtablegmatrixgagbgcgdeterminantfactoragfngdgegfgggigegfigffgfhgfdgfcgfbgfggfcopyrightarthurbelmontphilipyasskinepartmmathematictexauniversiteterminant"!ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalcanglcomputetweenvectorcallsequencparameterlistsamelengthdescriptcalculatbetweenequivallinalgexceptworkwithinsteadglelinearalgebraconjugatfalsoptionalwayselectcanalsowellpartpackagusednameonlyafterperformincommandaccesslongformexamplveccalugvgrtablegmatrixgvectorgcolumngpigcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitvectorangldotrtablsaveanythingggl"{ibmintelnthyperlinkhelpheadcouricouriertimetimesnormalbulletitemnormalfixedwidthdifferentialoperatorusingveccalcpackagcallsequencmakefunctoutevalfunctfncpointgradivarsouttyphessianleadingprincipalminordeterminantjacobianmatrixjacobianmatrixjacobiandetrminantjacobiandeterminantdivergencdivergencecurllaplacianscalarpotentialveccalcctorpotentialvectorpotentialparameternamelistvectornamrepresentindependvariablexpressatrixarrascalarmatrixvalufunctnestarrayevaluateddifferentiatformproducmakefunctatevaluatfieldarrowdefinvariablesfunctionsmustsquarcoordinattransformatctorfunctionjacobiandeterminantequaloptionalusedtypeoutputcurlchoicrowcolumnhessilistlistspecifidetermininputglobaloutputvectortypeoutputvectortypmatrixtypoutputmatrixtypdefaultdescriptthescommanddesignwithdefinitantiscalarincludparametrizsurfacmosthaveshortaliacommandsworkanydimenshowevnlyfunctionsali1crosspr"crosspro" crossprod" crossproduct"cs""""csg"ct """"ctg"ction"""ctor""""" ctorcalculu"ctorg "" ctorpotential ""cu""""culat"culu"cur"curl+"""""""curv3W"#" """"""""""curvatur"curveaccelerat"""" curvearcl"curvearclength""" curvebinormal""" curvecurvatur""" curveforget""""" curvejerk""" curvenormal"""curvenormalaccelerat"""curvespe""" curvetang"""curvetangentialaccel"curvetangentialaccelerat""""ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaltimesbulletitemanalysisurfacusingveccalcpackagcallsequencommandaliacommandparametercommlistbelowformvectorexpresswithparametersimplifsuchdescriptthesdesignperformeachshortonlyworkdimenscorrespondtsactionsurfacetangentstcalculattangsurfacenormalsncalculatsurfacenormallengthsnllengthnormalsurfaceareasaareasurfaceforgetsforgetclearremembtablabovusenameyoumustfirstexecutaliasvcaliavectexceptwillattemptmayplottusinplotwithparametricargumcommanusesremembtablespeedupcomputatafterfinishingavoidcluttermemordonecurveforgetexamplrtablegyxpvectorgcosgtgsingfrowgcossinrgceumatrixgtgfcolumngpigridrrrtrrgrtgkeufufhfgfhffhfngwfusingcosgflennlenngvaluagfagbgdgoperatorgarrowgfintgpigflngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvecurvrtablsaveanyth"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalwarnbulletitemfixedwidthdimensionalcoordinatconversusingveccalcpackagfunctioncylrectconvertcylindricalrectangularcoordinatesphsphericalsphericalsphericaloordinataliascanusedafterexecutvcaliacommandectcallsequencthetarhophihiparameterfirsthorizontalcoordinatsecondverticalaxispositupwardrelatedrighthandruletacoordinatperpendiculardistancanglmeasurradiancounterclockwissamecautsystemleftradialoriginpolarpositivdescriptthformulacosxgrgcosgthetagfsinygsingetagfzgfnthernorestrictvalucylindricalcoordinatesarctanthetagquadrantivsqrtsqrtgygfpiarctangpigfiiiiintheresultcylindricalrangpigfunctwithargumentdesignproducexactneedrhogphigfzghigfessqrtgphigcylindricalgfarccoarccosghethesreturnfloatpointdecimalnumbercontainanyfunctionspartnameonlyperformalwayaccesslongformexamplbgf"ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadcouriernormalwarnbulletitemfunctveccalccroscalculatproductvectoroperatorroductcallsequencesouttypautionmustspacafterelsewilthinksubsequletterpartnameparameterlistachwithelementoptionaltypevectorrowcolumndescriptdimensionalvectorsreturnanswparametspecificonverthaveheotherwimatchequivallinalgcrossprodexceptworkinsteadistslinearalgebracrossproductcanalsowellpackagusedonlyperformcommandalwayaccesslongformexamplugvgagbgcgcgfbgfagftorrtableggrmatrixgvectorgcolumngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversiteccalclinalcrossproductcrossproductdotneutralprecedencoperators"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 "Aibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriercouriernormalbulletitemfixedwidthnormaltypeveccalcpackagenvironmvariabloutputvectortypsetsomevectorvalufunctionoutputmatrixtypsommatrixcallsequencoutputvectortypvectortypmatrixtypparametrslistrowcolumnlistlistdescriptspecifgradifunctunlesoverriddenoptionallastinitialnotesynonymutputhessianunlessoverriddenthesbutcanresetanyafterperformcommandwithexamplvcaliafgxgygzgoperatorgarrowgfwhattyprtablegvectorgfbfrowgoutputvectortypeglistgarrowgfmatrixgfdfcolumnggmfcffnffiffefoffrfoutputmatrixtypeglistlistgoperatorgfhffkffgfcopyrightarthurbelmontphilipasskindepartmmathematictexauniversitalsoveccalcdiffoprtablsaveanythingggloperatorgoperatorg"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalwarnbulletitemtimescommandveccalcmakefunctmakefunctusingpackagoperatorfunctionaliacanusedafterexecutvcaliamfcallsequencoutveccalccauttheremayneedspacelsehinksubsequsymbolpartnameutexpresslistarrayouenclosparenthesparametervectorrepresentindependvariablmatrixrepresntingscalarvalunestarraydescriptdefinvaluedmorevariablarrowformproducefunctionsimilarothertypstructuruseevalfunctevaluatrrayveccalcthesonlyafterperformwithalwayaccesslongcommanexamplfgftgoperatorgarrowgfcurvfgseveraldensitsinlnxgygzgoperatorgsinglngvariablesfieldcoordinattransformatparametricsurfacrtablegmatrixgugvgvectorgcolumngrtableghefafptptgkwagbgcgrtablegcgfgfcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopneutralprecedenc"! ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemtimesdashfixedwidthfixedintroductveccalcpackagverspossiblpagelistatbottomcallsequenccommandargsdescriptcollectiondesignsimplifcalculatarisvectorcalculuproblemloadackagexecutwithpackageautomaticalstudplotmanycommandshaveshortaliasbecomavailablaftervcaliadividintoseveralgroupthesbelowfollowaliaparentheseachhyperlinkenvironmsettingoutputvectortypoutputtypoutputmatrixtypmakeevaluatscalarvectorfunctionmakefunctmfevalfunctionefperformlinearalgebracomputatdotlengthlenanglcrossimplifyvecdeterminantdetleadingprincipalminordeterminantleadingprincipalminordeterminantlpmdchangangularmeasurcoordinatangleconverscoordconverscoordconversiondegrangleconversradangleconversionpolarrectolarcylsphcoordconveranalysecurvcurvevelocitcvcurveacceleratcaejerkcjcurvetangctcurvenormalcncurvebinormalcb+curvespecscurvearclngthclcurvecurvaturcketorscurvetangentialacceleratcatcurvenormalacceleratcancurveforgetcforgetcommandsanalyssurfacsurfacetangstsurfacenormalsnsurfacenormallengthsnlsurfaceareasasurfaceforgetsforgetdifferentialoperatdiffopmultimaxmingradigradhessianhessdivergencdivcurllaplacianlapjacobianmatrixjacjacobiandeterminantjdetscalarpotentialscalarpotentialspotvectorpotentialvectorpotntialvpotntegralmultipleintmuintmultipleimultipleintlineintscalarinertlislineintscalarlineintvectorinertlineintvectorlivneintvectorsurfaceintscalarinertsurfaceintscalarsissurfaceintvectorinertintvectorsivsurfaceintvectsurfaceintvectorinternalunavailablusergetvectypmapfuncmapunappgetvarsnochkweremodifiontomatricarraymappedfunctlimitdiffintvaluexamplproductusewithagrtablegmatrixgxgygzgvectorgcolumngbgmatrixgectorgygfzgfacknowledgementusedextensivethroughouttextmultivariablecalclabarthurbelmont,philipyasskinpublishbrookcoleseveraleditiondiffertitloriginalwrittenreleasbelmontorganizjamewarrenfirstdavidarnoldconvertkenparkjaredteslomodulupdatwereelpsignificantcompatiblwitnewmatrixtypechadwellingtonkristaristchrihaagethanmcconnelallisondenbleykndjeffrecopyrightdepartmmathematicstexauniversitallrightreservalsolibnamiffopsioncoordconversanyyouveccalcabovreferencherepicspointthemjustclickmaioutputvectortpeevalfunctmappedfunctionvaluecrosslinearalgebraleadingprincipalminordeterminantangleconverspolarectfrenetcurvejerkcurvearclengthcurvetorssurfacetangentsurfacenormallengthsurfacearsurfaceforgetdiffopsdivergncejacobianmatrixctorpotentialmuilineintscalarinertlineintvectorinertsurfaceintvectorimitdiffintvaluexamplproductusewithagrtablegmatrixgxgygzgvectorgcolumngbgmatrixgectorgygfzgfacknowledgementusedextensivethroughouttextmultivariablecalclabarthurbelmont,MappedFunctions'Mathematics/Packages/VecCalc/Commands/D MultiMaxMin(Mathematics/Packages/VecCalc/MultiMaxMin Multipleint(Mathematics/Packages/VecCalc/Multipleint Multipleint1Mathematics/Packages/VecCalc/Commands/Multipleint Multipleint1Mathematics/Packages/VecCalc/Commands/multipleint Multipleint*Mathematics/Packages/VecCalc/Aliases/Muint Multipleint*Mathematics/Packages/VecCalc/Aliases/muint OutputTypes(Mathematics/Packages/VecCalc/OutputTypes OutputTypes6Mathematics/Packages/VecCalc/Commands/OutputVectorType OutputTypes6Mathematics/Packages/VecCalc/Commands/OutputMatrixTypeScalarPotential,Mathematics/Packages/VecCalc/ScalarPotentialScalarPotential5Mathematics/Packages/VecCalc/Commands/ScalarPotentialScalarPotential)Mathematics/Packages/VecCalc/Aliases/SPotAngle"AngleConversion"CoordConversion2D"CoordConversion3D"Cross"Curl"Curve" CurveForget" Determinant"Diffops" Divergence"Dot"Gradient"Hessian"JacobianDeterminant"JacobianMatrix" Laplacian"!LeadingPrincipalMinorDeterminants"Length" LineIntScalar" LineIntVector" MakeFunction"MappedFunctions" MultiMaxMin" Multipleint" OutputTypes"ScalarPotential"Surface" SurfaceForget"SurfaceIntScalar"SurfaceIntVector"VCalias"VecCalc"VectorPotential" evalFunction" simplifyvec""'{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b";${VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 p"=0{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 22 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Fixed Widt h" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "" 256 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 03 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 256 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 04 0 0 0 0 -1 0 }{PSTYLE "" 256 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 256 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 280 1 {CSTYLE5 "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Command:" }{TEXT -1 1 " " }{TEXT 257 19 "VecCalc[VCalias] - " }{TEXT -1 38 "Sets aliases for som e commands in the " }{TEXT 258 7 "VecCalc" }{TEXT -1 9 " package." }} {PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequence: " }}{PARA 0 "" 0 "" {TEXT 256 10 " VCalias" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Desc ription:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 54 "This command defines aliases for some commands in the " }{TEXT 259 7 "VecCalc" } {TEXT -1 38 " package. This is done using Maple's " }{HYPERLNK 17 "al ias" 2 "alias" "" }{TEXT -1 56 " command. So the output includes all \+ previous aliases. " }}{PARA 15 "" 0 "" {TEXT -1 48 "The aliases (with \+ links to their help pages) are" }}{PARA 260 "" 0 "" {TEXT -1 12 " MF \+ = " }{HYPERLNK 17 "MakeFunction" 2 "MakeFunction" "" }{TEXT -1 21 " EF = " }{HYPERLNK 17 "evalFunction" 2 "evalFunctio n" "" }{TEXT -1 166 " " }}{PARA 261 "" 0 "" {TEXT -1 12 " Det = " }{HYPERLNK 17 "Determinant" 2 "LinearAlgebra[Determinan t]" "" }{TEXT -1 50 " \+ " }}{PARA 262 "" 0 "" {TEXT -1 12 " LPMD = " }{HYPERLNK 17 "Leadi ngPrincipalMinorDeterminants" 2 "LeadingPrincipalMinorDeterminants" " " }{TEXT -1 28 " " }}{PARA 263 "" 0 "" {TEXT -1 12 " d2r = " }{HYPERLNK 17 "deg2rad" 2 "deg2rad" "" } {TEXT -1 26 " r2d = " }{HYPERLNK 17 "rad2deg" 2 "de g2rad" "" }{TEXT -1 21 " " }}{PARA 264 "" 0 "" {TEXT -1 12 " p2r = " }{HYPERLNK 17 "polar2rect" 2 "CoordConversi on2D" "" }{TEXT -1 23 " r2p = " }{HYPERLNK 17 "rect2po lar" 2 "CoordConversion2D" "" }{TEXT -1 18 " " }} {PARA 265 "" 0 "" {TEXT -1 12 " c2r = " }{HYPERLNK 17 "cyl2rect" 2 "CoordConversion3D" "" }{TEXT -1 25 " r2c = " } {HYPERLNK 17 "rect2cyl" 2 "CoordConversion3D" "" }{TEXT -1 720 " \+ " }}{PARA 266 "" 0 "" {TEXT -1 12 " s2r = " } {HYPERLNK 17 "sph2rect" 2 "CoordConversion3D" "" }{TEXT -1 25 " \+ r2s = " }{HYPERLNK 17 "rect2sph" 2 "CoordConversion3D" "" }{TEXT -1 20 " " }}{PARA 267 "" 0 "" {TEXT -1 12 " \+ s2c = " }{HYPERLNK 17 "sph2cyl" 2 "CoordConversion3D" "" }{TEXT -1 26 " c2s = " }{HYPERLNK 17 "cyl2sph" 2 "CoordCon version3D" "" }{TEXT -1 21 " " }}{PARA 268 "" 0 " " {TEXT -1 12 " Cv = " }{HYPERLNK 17 "CurveVelocity" 2 "Curve" " " }{TEXT -1 20 " CT = " }{HYPERLNK 17 "CurveTangent" 2 " Curve" "" }{TEXT -1 16 " " }}{PARA 269 "" 0 "" {TEXT -1 12 " Ca = " }{HYPERLNK 17 "CurveAcceleration" 2 "Curve" "" } {TEXT -1 16 " CN = " }{HYPERLNK 17 "CurveNormal" 2 "Curve" " " }{TEXT -1 17 " " }}{PARA 270 "" 0 "" {TEXT -1 12 " \+ Cj = " }{HYPERLNK 17 "CurveJerk" 2 "Curve" "" }{TEXT -1 24 " \+ CB 8 = " }{HYPERLNK 17 "CurveBinormal" 2 "Curve" "" } {TEXT -1 15 " " }}{PARA 271 "" 0 "" {TEXT -1 12 " Cs \+ = " }{HYPERLNK 17 "CurveSpeed" 2 "Curve" "" }{TEXT -1 23 " \+ CL = " }{HYPERLNK 17 "CurveArcLength" 2 "Curve" "" }{TEXT -1 14 " " }}{PARA 272 "" 0 "" {TEXT -1 12 " Ck = " } {HYPERLNK 17 "CurveCurvature" 2 "Curve" "" }{TEXT -1 19 " CaT \+ = " }{HYPERLNK 17 "CurveTangentialAcceleration" 2 "Curve" "" } {TEXT -1 1 " " }}{PARA 273 "" 0 "" {TEXT -1 12 " Ct = " } {HYPERLNK 17 "CurveTorsion" 2 "Curve" "" }{TEXT -1 21 " CaN \+ = " }{HYPERLNK 17 "CurveNormalAcceleration" 2 "Curve" "" }{TEXT -1 5 " " }}{PARA 274 "" 0 "" {TEXT -1 12 " Cforget = " } {HYPERLNK 17 "CurveForget" 2 "curve_forget" "" }{TEXT -1 50 " \+ " }}{PARA 275 "" 0 "" {TEXT -1 12 " ST = " }{HYPERLNK 17 "SurfaceTangents" 2 "Surface" "" } {TEXT -1 18 " SN = " }{HYPERLNK 17 "SurfaceNormal" 92 "Surf ace" "" }{TEXT -1 15 " " }}{PARA 276 "" 0 "" {TEXT -1 12 " SNL = " }{HYPERLNK 17 "SurfaceNormalLength" 2 "Surface" "" } {TEXT -1 14 " SA = " }{HYPERLNK 17 "SurfaceArea" 2 "Surface" " " }{TEXT -1 17 " " }}{PARA 277 "" 0 "" {TEXT -1 12 " \+ Sforget = " }{HYPERLNK 17 "SurfaceForget" 2 "SurfaceForget" "" }{TEXT -1 48 " " }}{PARA 278 " " 0 "" {TEXT -1 12 " Grad = " }{HYPERLNK 17 "Gradient" 2 "Gradient " "" }{TEXT -1 25 " Hess = " }{HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 21 " " }}{PARA 17 "" 0 " " {TEXT -1 12 " Div = " }{HYPERLNK 17 "Divergence" 2 "Divergence " "" }{TEXT -1 23 " Lap = " }{HYPERLNK 17 "Laplacian" 2 "Laplacian" "" }{TEXT -1 19 " " }}{PARA 17 "" 0 " " {TEXT -1 12 " Jac = " }{HYPERLNK 17 "JacobianMatrix" 2 "Jacobia nMatrix" "" }{TEXT -1 19 " JDet = " }{HYPERLNK 17 "Jacobian Determinant" 2 "Jacob:ianDeterminant" "" }{TEXT -1 9 " " }} {PARA 17 "" 0 "" {TEXT -1 12 " SPot = " }{HYPERLNK 17 "ScalarPoten tial" 2 "ScalarPotential" "" }{TEXT -1 18 " VPot = " } {HYPERLNK 17 "VectorPotential" 2 "VectorPotential" "" }{TEXT -1 13 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 12 " Muint = " } {HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 22 " \+ muint = " }{HYPERLNK 17 "multipleint" 2 "Multipleint" "" }{TEXT -1 17 " " }}{PARA 258 "" 0 "" {TEXT -1 12 " Lis = " }{HYPERLNK 17 "InertLineIntScalar" 2 "LineIntScalar" "" }{TEXT -1 15 " lis = " }{HYPERLNK 17 "LineIntScalar" 2 "LineIntScalar" " " }{TEXT -1 15 " " }}{PARA 259 "" 0 "" {TEXT -1 12 " Li v = " }{HYPERLNK 17 "InertLineIntVector" 2 "LineIntVector" "" } {TEXT -1 15 " liv = " }{HYPERLNK 17 "LineIntVector" 2 "LineInt Vector" "" }{TEXT -1 15 " " }}{PARA 279 "" 0 "" {TEXT -1 12 " Sis = " }{HYPERLNK 17 "InertSurfaceIntScalar" 2 "SurfaceI; ntScalar" "" }{TEXT -1 12 " sis = " }{HYPERLNK 17 "SurfaceIntScal ar" 2 "SurfaceIntScalar" "" }{TEXT -1 12 " " }}{PARA 280 " " 0 "" {TEXT -1 12 " Siv = " }{HYPERLNK 17 "InertSurfaceIntVector " 2 "SurfaceIntVector" "" }{TEXT -1 12 " siv = " }{HYPERLNK 17 "S urfaceIntVector" 2 "SurfaceIntVector" "" }{TEXT -1 12 " " } }{PARA 15 "" 0 "" {TEXT -1 28 "This command is part of the " }{TEXT 260 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 261 7 "VCalias" }{TEXT -1 35 " only after performing the com mand " }{TEXT 262 13 "with(VecCalc)" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 " Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6U%&PointG%#MFG%#EFG%$DetG%%LPMDG%$d2rG%$r2dG%$p2rG%$r2pG %$c2rG%$r2cG%$s2rG%$r2sG%$s2cG%$c2sG%#CvG%#CaG%#CjG%#CsG%#CLG%#CTG%#CN G%#CBG%#CkG%#CtG%$CaTG%$CaNG%(CforgetG%#STG%#SNG%$SNLG%#SAG%(SforgetG% %GradG%%HessG%$DivG%$LapG%$Ja<cG%%JDetG%%SPotG%%VPotG%&MuintG%&muintG%$ LisG%$lisG%$LivG%$livG%$SisG%$sisG%$SivG%$sivG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "f:=MF([x,y,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^ 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%f*6%%\"xG%\"yG%\"zG6\" 6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9& \"\"%F4F4F+F+F+f*F'F+F,F+*&)F7F3F4F;F4F+F+F+f*F'F+F,F+*&F1F4)F;F8F4F+F +F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f:=MakeFunction([x,y ,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\" #\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+f*F'F+F,F+*&)F7F3F4 F;F4F+F+F+f*F'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 8 "See Also" }{TEXT -1 2 ": " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "alia=s" 2 "alia s" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } " {MPLTEXT 1 0 50 "f:=MF([x,y,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^ 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%f*6%%\"xG%\"yG%\"zG6\" 6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9& \"\"%F4F4F+F+F+f*F'F+F,F+*&)F7F3F4F;F4F+F+F+f*F'F+F,F+*&F1F4)F;F8F4F+F +F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f:=MakeFunction([x,y ,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\" #\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+f*F'F+F,F+*&)F7F3F4 F;F4F+F+F+f*F'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 8 "See Also" }{TEXT -1 2 ": " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "alia="1"E"S"Z"/"0"_"""'"""""`""""n""2"W"f"u"~""""""""""ediat"edition"edo"edt"ef """""""efg"efunct"eg"""egf"egrat"eint""" eintscalar"eintvectorinert"eith ""ejerk"eld"element"eliminat"ellips"elmont"elp"else"""ement"en"ence"enclo"enclos"ential ""entr"environm ""ep"epartm"epend"eps"ept"eqs"eqsg"equal ""alias;&""""""""""""""all/"""""""""""alling"allison"allvalu"along"als"also.""""""""""""""""""""""""""""""""""""alternat"always""""""""""""""""""""""""""""ameter"""ametriz"anal ""analys"analysi """" andeterminant" functionalit"fy"fyvec"ge"genc"get" getvectyp"gf#""""""""gg""""ggf""""""give"gl;"""""""""""""""gle"global"""gm"gn"gnitud"gp"gprincipalminordeterminant"gr"""grad""""""gradg"gradi/;""""" """""""KibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemimesfixedwidthwidtcommandveccalcvcaliasetsaliassompackagcallsequencdescriptdefinsomedoneusingaliasaliaincludallpreviouwithlinktheirpagemfmakefunctefevalfunctevalfunctiodetdeterminantlinearalgebradeterminanlpmdleadingprincipalminordeterminantleadingprincipalminordeterminantdegraddepolarrectcoordconversipolarcoordconverscylsphcoordconverscvcurvevelocitcurvctcurvetangcacurveacceleratcncurvenormalcjcurvejerkcbcurvebinormalcscurvespeclcurvearclengthckcurvecurvaturcatcurvetangentialacceleratcurvetorscancurvenormalacceleratcforgetcurveforgetforgetstsurfacetangentsurfacsnsurfacenormalsurfacesnlsurfacenormallengthsasurfaceareasforgetsurfaceforgetgradgradihesshessiandivdivergenclaplaplacianjacjacobianmatrixjacobianmatrixjdetjacobianjacobiandeterminantspotscalarpotentialscalarpotentialvpotvectorpotenCtialmuintmultipleintlisinertlineintscalarlineintscalarliinertlineintvectorlineintvectorlivlineintvectorsisinertsurfaceintscalarsurfaceintscalarsurfaceintscalarsurfaceintscalarsivinertsurfaceintvectorsurfaceintvectorurfaceintvectorpartusedformonlyafterperformcommandexamplpointgmfgefgdetglpmdgrgdgpgcgsgcvgcagcjgcsgclgctgcbgckgcatgcangcforgetgstgsngsnlgsagsforgetggradghessgdivglapgjacgjdetgspotgvpotgmuintglisglivgsisgsivgfgxgygzgoperatorgarrowgfcopyrightarthurbelmontphilipasskindepartmmathematictexauniversitalsovespeclcurvearclengthckcurvecurvaturcatcurvetangentialacceleratcurvetorscancurvenormalacceleratcforgetcurveforgetforgetstsurfacetangentsurfacsnsurfacenormalsurfacesnlsurfacenormallengthsasurfaceareasforgetsurfaceforgetgradgradihesshessiandivdivergenclaplaplacianjacjacobianmatrixjacobianmatrixjdetjacobianjacobiandeterminantspotscalarpotentialscalarpotentialvpotvectorpotenC*"B"(""m")"""O"!"&" """]"%""z"$"^""""""{"""""""X"""""${VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1F 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 10 0 0 0 0 0 0 0 0 0 0 0G 0 0 1 }{CSTYLE "" -1 283 "Cour ier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Nor mal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bul let Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Cour Hier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 21 " \+ Output Types in the " }{TEXT 261 7 "VecCalc" }{TEXT -1 8 " Package" }} {PARA 0 "" 0 "" {TEXT 26 22 "Environment Variables:" }}{PARA 0 "" 0 " " {TEXT 256 22 " OutputVectorType - " }{TEXT 262 53 "Set the output \+ type for some vector valued functions." }}{PARA 0 "" 0 "" {TEXT 257 22 " OutputMatrixType - " }{TEXT 263 53 "Set the output type for som e matrix valued functions." }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Seq uences:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 264 17 "OutputVec torType " }{TEXT -1 3 ":= " }{TEXT 265 10 "vectortype" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 266 16 "OutputMatrixType" }{TEXT -1 4 " : = " }{TEXT 267 10 "matrixtype" }}{PARA 0 "" 0 "" {TEXT 26 11 "Paramete rs:" }}{PARA 0 "" 0 "" {TEXT 258 49 " vectortype - 'list', 'Vector ', 'Vector[row]'" }{TEXT 269 4 " or " }{TEXT 270 17 "'Vector[column]'. " }{TEXT -1 1 "\n" }{TEXT I268 28 " matrixtype - 'listlist'" } {TEXT 271 4 " or " }{TEXT 272 8 "'Matrix'" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT 273 16 "OutputVectorType" }{TEXT -1 35 " specifies the output type for the " }{TEXT 274 8 "Gradient" }{TEXT -1 71 " function unless it is ov erridden by an optional last parameter in the " }{TEXT 275 8 "Gradient " }{TEXT -1 17 " function call. " }{TEXT 276 16 "OutputVectorType" } {TEXT -1 21 " is initially set to " }{TEXT 277 13 "'Vector[row]'" } {TEXT 278 9 ". Note: " }{TEXT 285 6 "Vector" }{TEXT 286 18 " is a syn onym for " }{TEXT 287 14 "Vector[column]" }{TEXT 288 1 "." }}{PARA 15 "" 0 "" {TEXT 279 16 "OutputMatrixType" }{TEXT -1 35 " specifies the o utput type for the " }{TEXT 283 7 "Hessian" }{TEXT -1 71 " function un less it is overridden by an optional last parameter in the " }{TEXT 284 7 "Hessian" }{TEXT -1 17 " function call. " }{TEXT 280 16 "Output MatrixType" }{TEXT -1 21 " is initially set to " }{TEXT 281 8 "'MJatrix '" }{TEXT 282 1 "." }}{PARA 15 "" 0 "" {TEXT -1 53 "These environment \+ variables are initially set by the " }{TEXT 259 7 "VecCalc" }{TEXT -1 65 " package, but can be reset any time after performing the command \+ " }{TEXT 260 13 "with(VecCalc)" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=(x,y,z)->x^2*y^3*z^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF**()9$\"\"#\"\"\")9%\"\"$F 2)9&\"\"%F2F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Gradie nt(f); whattype(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\") W/o=-%'VECTORG6#7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF0,$**\" \"#\"\"\"9$F7)9%\"\"$F7)9&\"\"%F7F7F0F0F0f*F,F0F1F0,$**F;F7)F8F6F7)F:F 6F7FF7FBF7F9F7)F=F;F7F7F0F0F0&%'VectorG6#%$ rowG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MKPLTEXT 1 0 25 "OutputVectorType:='list';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1OutputVectorTypeG%%listG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Gradient(f); whattype(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrow GF),$**\"\"#\"\"\"9$F0)9%\"\"$F0)9&\"\"%F0F0F)F)F)f*F%F)F*F),$**F4F0)F 1F/F0)F3F/F0F5F0F0F)F)F)f*F%F)F*F),$**F7F0F;F0F2F0)F6F4F0F0F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%listG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Gradient(f, Vector ); whattype(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6%\")W1o=-%'MATRIXG6#7%7#f*6%%\"xG%\"yG%\"z G6\"6$%)operatorG%&arrowGF1,$**\"\"#\"\"\"9$F8)9%\"\"$F8)9&\"\"%F8F8F1 F1F17#f*F-F1F2F1,$**FF " 0 "" {MPLTEXT 1 0 24 "Hessian(f); whattype(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!GM(=-%'MATRIXG6#7%7%f*6%%\"xG%\"yG%\"zGL 6\"6$%)operatorG%&arrowGF1,$*(\"\"#\"\"\")9%\"\"$F8)9&\"\"%F8F8F1F1F1f *F-F1F2F1,$**\"\"'F89$F8)F:F7F8F " 0 "" {MPLTEXT 1 0 27 "OutputMatrixType:=listlist; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1OutputMatrixTypeG%)listlistG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Hessian(f); whattype(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%7%f*6%%\"xG%\"yG%\"zG6\"6$%)operat orG%&arrowGF*,$*(\"\"#\"\"\")9%\"\"$F1)9&\"\"%F1F1F*F*F*f*F&F*F+F*,$** \"\"'F19$F1)F3F0F1F5F1F1F*F*F*f*F&F*F+F*,$**\"\")F1FFHf*F&F*F+F*,$**FKF1FGF1F2F1)F6F0F1F1F*F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%listG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995M-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "Ve cCalc" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" } {TEXT 26 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT 26 2 " , " }{HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 1 "." }}}}{MARK " 0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 18680444 18680644 18734280 }{RTABLE M7R0 I5RTABLE_SAVE/18680444X*%)anythingG6"6"[gl!$%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,$*(9$"""9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(F2F8F4F8F6F7F5F,F,F,f*F (F,F-F,,$*(F2F8F4F5F6F5F7F,F,F,F, } {RTABLE M7R0 I5RTABLE_SAVE/18680644X*%)anythingG6"6"[gl!#%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,$*(9$"""9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(F2F8F4F8F6F7F5F,F,F,f*F (F,F-F,,$*(F2F8F4F5F6F5F7F,F,F,F, } {RTABLE M7R0 I5RTABLE_SAVE/18734280X,%)anythingG6"6"[gl!"%!!!#*"$"$f*6%%"xG%"yG%"zG6"6$%)ope rNatorG%&arrowGF,,$*&9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(9$"""F2F6F4F5""'F,F,F,f*F( F,F-F,,$*(F:F;F2F3F4F3"")F,F,F,F7f*F(F,F-F,,$*(F:F6F2F;F4F5F " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u:=[1,2,3]; v:=[x,y,z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrt(2)*u + 3*v; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"## \"\"\"F%7%F'F%\"\"$F'F'7%,$*&F)F'%\"xGF'F',$*&F)F'%\"yGF'F',$*&F)F'%\" zGF'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 V"simplifyvec(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$\"\"##\"\"\"F&F(*&\"\"$F(%\" xGF(F(,&*&F&F(F&F'F(*&F*F(%\"yGF(F(,&*&F*F(F&F'F(*&F*F(%\"zGF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=<,>; B:=<<1|2>, <3|4>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")cr'H\" -%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")_Q*G\"-%'MATRIXG6#7$7$\"\"\"\"\"#7 $\"\"$\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A*x+ B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%'RTABLEG6%\")cr 'H\"-%'MATRIXG6#7$7$%\"aG%\"bG7$%\"cG%\"dG%'MatrixGF&F&-F(6%\")_Q*G\"- F,6#7$7$F&\"\"#7$\"\"$\"\"%F5F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplifyvec(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6 %\")_Z39-%'MATRIXG6#7$7$,&\"\"\"F-*&%\"xGF-%\"aGF-F-,&\"\"#F-*&F/F-%\" bGF-F-7$,&\"\"$F-*&F/F-%\"cGF-F-,&\"\"%F-*&F/F-%\"dGF-F-%'MatrixG" }}} {EXCHG }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur BelmWonte and Philip B. Yasskin\n Department of Mathematic s, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "convert" 2 "convert" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12967156 12893852 14084752 }{RTABLE M7R0 I5RTABLE_SAVE/12967156X,%)anythingG6"6"[gl!"%!!!#%"#"#%"aG%"cG%"bG%"dGF& } {RTABLE M7R0 I5RTABLE_SAVE/12893852X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""$""#""%F& } {RTABLE M7R0 I5RTABLE_SAVE/14084752X,%)anythingG6"6"[gl!"%!!!#%"#"#,&"""F(*&%"xGF(%"aGF(F(,& ""$F(*&F*F(%"cGF(F(,&""#F(*&F*F(%"bGF(F(,&""%F(*&F*F(%"dGF(F(F& } -%'RTABLEG6 %\")_Z39-%'MATRIXG6#7$7$,&\"\"\"F-*&%\"xGF-%\"aGF-F-,&\"\"#F-*&F/F-%\" bGF-F-7$,&\"\"$F-*&F/F-%\"cGF-F-,&\"\"%F-*&F/F-%\"dGF-F-%'MatrixG" }}} {EXCHG }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur BelmW"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfixedwidthfunctionveccalcmultipleintdisplayinertmulitiplintegralveccalccomputmultiplbothcommandcandisplaintermediatstepsaliasaliasesusedafterexecutvcaliacaliamuintcallsequencxnmultipleinstepintparameterexpressintegrandeachnamerangspecifvariablintegratoptionalrangerequirparametindicatdescriptfirstargumfollowargumentntegratincludnumericalappearordertheyevaluatdifferentialusingvaluevalfcalculatwithoutwhilallmultipleintreturnyoudoneedquotaroundassignfunctionalitstuddoubleinttripleinttripleintpackagbutalsoworkhighdimensionalintegralsthespartonlyperformwithfunctalwayaccesslongformsedvcaliasexamplesmuinintgxgygzgevalgxgfevalgcopyrightarthurelmontphilipyasskindepartmmathematictexauniversitdoubleinjacobiandeterminantlineintscalarlineintvectorsurfaceintscalarsurfaceintvctorsurfacclud"clutter ""cn"""co"""""cobian"cole"collec"colu ""colum"column# """"""""columngC""""""""""""""""columngf"com""""""comm""""comma ""comman""""command"""""""""""""""""" " " """""""""""""""""b {VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Fixed Width" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Fixed Wi dth" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 17 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 40 " \+ Mapped Functions in the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:"[ }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 25 " Limit VecCalc[Limit]" }}{PARA 257 "" 0 "" {TEXT -1 25 " lim it VecCalc[limit]" }}{PARA 257 "" 0 "" {TEXT -1 24 " Diff VecCa lc[Diff]" }}{PARA 257 "" 0 "" {TEXT -1 24 " diff VecCalc[diff]" } }{PARA 257 "" 0 "" {TEXT -1 23 " Int VecCalc[Int]" }}{PARA 257 " " 0 "" {TEXT -1 23 " int VecCalc[int]" }}{PARA 257 "" 0 "" {TEXT -1 25 " value VecCalc[value]" }}{PARA 257 "" 0 "" {TEXT -1 21 " D VecCalc[D]" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Des cription: " }}{PARA 15 "" 0 "" {TEXT -1 187 "These commands are exactl y like the usual Maple commands of the same names, except that whan th e input is a list, Vector, Matrix or Array, they are applied to each c omponent of the input." }}{PARA 15 "" 0 "" {TEXT -1 49 "To use the com mand names, you must first execute " }{TEXT 256 13 "with(VecCalc)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "- Copyright 200 3 by Jeffrey Yasskin and Philip B. Yasskin\n Departme\nt of Mathem atics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also: " }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Limit" 2 "Limit" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "limit" 2 "limit" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diff" 2 "Diff" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "diff" 2 "diff" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "D" 2 "D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value." 2 "value" "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ple commands of the same names, except that whan th e input is a list, Vector, Matrix or Array, they are applied to each c omponent of the input." }}{PARA 15 "" 0 "" {TEXT -1 49 "To use the com mand names, you must first execute " }{TEXT 256 13 "with(VecCalc)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "- Copyright 200 3 by Jeffrey Yasskin and Philip B. Yasskin\n Departme\"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaltimesbulletitemfixedwidthdimensionalcoordinatconversusingveccalcpackagfunctionpolarrectconvertrectangularcoordinatealiascanusedafterexecutvcaliacommandcallingsequencolarthetaparameterhorizontalcoordinatepositrightverticalupwardcoordinatradiadistancoriginanglmeasurradiancounterclockwisositaxisdescriptthformulacosxgrgcosgthetagfsinygsingnthernorestrictvaluarctanthetagarctangquadrantivsqrtsqrtgygfpipigfiiiiintheresultcoordinatesrangpigfunctwithargumentdesignproducexactlyneedthesunctionreturnfloatpointdecimalnumbercontainnypartnameonlyperformalwayaccesslongformxamplrecagbgfsingfrtablegvectorgarctangrowgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconversangleconversangleconversi"Zibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemtimesfunctveccalcsurfaceforgetclearremembertablsurfacanalysialiacanusedafterexecutvcaliacommandsforgetcallsequenccforgetparametereachformlistvectorexpresswithdescriptpackagperformsuchsurfacetangentsurfacenormaluseremembstortheirresultutsdowncomputotherthesallartlyeccalcalwayaccesslongonlyperforminexamplsincosrgrtablegoxlvectorgtgsingcosgfrowgsnoxpcosgsingfcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalso"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }":{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 )Mathematics/Packages/VecCalc/Determinant*Mathematics/Packages/VecCalc/MakeFunction%Mathematics/Packages/VecCalc/VCalias)Mathematics/Packages/VecCalc/Aliases/CaN)Mathematics/Packages/VecCalc/Aliases/Det(Mathematics/Packages/VecCalc/Aliases/SA*Mathematics/Packages/VecCalc/Aliases/VPot )Mathematics/Packages/VecCalc/Commands/&.8Mathematics/Packages/VecCalc/Commands/CurveAcceleration 1Mathematics/Packages/VecCalc/Commands/CurveSpeed +Mathematics/Packages/VecCalc/Commands/Diff0Mathematics/Packages/VecCalc/Commands/Laplacian7Mathematics/Packages/VecCalc/Commands/OutputMatrixType6Mathematics/Packages/VecCalc/Commands/SurfaceTangents*Mathematics/Packages/VecCalc/Commands/int} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Command:" }{TEXT -1 1 " " }{TEXT 261 24 "VecCalc[MakeFunctcion] - " }{TEXT -1 41 "Make a Function using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 9 "Operator:" } {TEXT -1 1 " " }{TEXT 270 15 "VecCalc[&->] - " }{TEXT -1 41 "Make a Fu nction using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias: " }{TEXT -1 48 " - The alias can be used after execution of the " } {HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 256 22 " MF = MakeFunction" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 81 " MakeFunction(in,out) in &-> out MF(in,out) VecC alc[MakeFunction](in,out)" }}{PARA 7 "" 0 "" {TEXT -1 231 "CAUTION: Th ere may need to be a space after the operator &->, or else Maple may t hink subsequent symbols are part of the name of the operator. If the o ut expression is not a list or Array, you may need to enclose it in pa rentheses." }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT 258 11 " in - " }{TEXT -1 88 "a namde or a list of names or a Vector of names representing the independent var iable(s)." }}{PARA 0 "" 0 "" {TEXT 259 11 " out - " }{TEXT -1 134 "an expression or list, Vector, Matrix or Array of expressions represe nting the scalar, vector, matrix or array value of the function,\n" } {TEXT 260 11 " " }{TEXT -1 42 "OR nested lists and Arrays of expressions." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 268 12 "MakeFunction" }{TEXT -1 153 " defines a scalar- or list- or vector- or matrix- or array-val ued function of one or more variables in arrow form. If the function \+ is list-valued, then " }{TEXT 267 12 "MakeFunction" }{TEXT -1 97 " pro duces a list of arrow-defined functions. Similarly, for the other typ es of output structure." }}{PARA 15 "" 0 "" {TEXT -1 4 "Use " }{TEXT 269 12 "evalFunction" }{TEXT -1 56 " to evaluate Vector-, Matrix- or A rray-valued functions." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function \+ " }{TEXT 272 12 "MakeeFunction" }{TEXT -1 18 " and the operator " } {TEXT 271 3 "&->" }{TEXT -1 17 " are part of the " }{TEXT 262 7 "VecCa lc" }{TEXT -1 78 " package, and so can be used in these forms only aft er performing the command " }{TEXT 263 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 264 34 "with(VecCalc, MakeFunction, `&->`)" }{TEXT -1 55 ". The command can always be accessed in the long form " }{TEXT 265 21 "VecCalc[MakeFunction]" }{TEXT -1 13 ". The alias " }{TEXT 266 2 "MF" }{TEXT -1 47 " can be used only after performing the comman d " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{PARA 0 "" 0 "" {TEXT -1 34 "A scalar function of one variable:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "f:=MakeFunction(t,t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\" \"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 49 "A vector functiofn of one var iable (e.g. a curve):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= MF(t,[t,t^2,t^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%f*6#%\"t G6\"6$%)operatorG%&arrowGF)9$F)F)F)f*F'F)F*F)*$)F-\"\"#\"\"\"F)F)F)f*F 'F)F*F)*$)F-\"\"$F2F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 56 "A scalar fu nction of several variables (e.g. a density):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:= &-> (x^2*sin(t*y)+ln(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6&%\"xG%\"yG%\"zG%\"tG6\"6$%)operato rG%&arrowGF+,&*&)9$\"\"#\"\"\"-%$sinG6#*&9'F49%F4F4F4-%#lnG6#9&F4F+F+F +" }}}{PARA 0 "" 0 "" {TEXT -1 116 "A vector function of several varia bles (e.g. a vector field or a coordinate transformation or a parametr ic surface):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f:=MF(, <(u+v)/2, (u-v)/2>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'RTABL EG6%\"(C=e#-%'MATRIXG6#7$7#f*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF2,&* &#\"\"\"\"\"#F99%F9F9*&F8F99$F9F9F2F2F27#f*F/F2F3F2,&*&F8F9F=F9F9*&#F9 F:F9F;F9!\"\"Fg2F2F2&%'VectorG6#%'columnG" }}}{PARA 0 "" 0 "" {TEXT -1 39 "A Matrix function of several variables:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "f:=[x,y,t] &-> <,>;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'RTABLEG6%\"(;\"HE-%'MATRIXG6# 7$7%f*6%%\"xG%\"yG%\"tG6\"6$%)operatorG%&arrowGF3*$)9$\"\"#\"\"\"F3F3F 3f*F/F3F4F3,&F9F;9&F;F3F3F3f*F/F3F4F3,&9%F;*&)F9\"\"$F;)F>F:F;!\"\"F3F 3F37%f*F/F3F4F3F>F3F3F3f*F/F3F4F3FAF3F3F3F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "pt:=; f &@ pt;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#ptG-%'RTABLEG6%\")Kw::-%'MATRIXG6#7%7#%\"aG7#%\"bG 7#%\"cG&%'VectorG6#%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6%\")'\\4b\"-%'MATRIXG6#7$7%*$)%\"aG\"\"#\"\"\",&F.F0%\"cGF0,&%\"b GF0*&)F.\"\"$F0)F2F/F0!\"\"7%F2F4F,%'MatrixG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B . Yasskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{ThEXT -1 1 " " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalFunction" 2 "evalFunction" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface" 2 "Surface" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "operator" 2 "operator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "neutral" 2 "neutral" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operators[ precedence]" 2 "operators[precedence]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ")Kw::-%'MATRIXG6#7%7#%\"aG7#%\"bG 7#%\"cG&%'VectorG6#%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6%\")'\\4b\"-%'MATRIXG6#7$7%*$)%\"aG\"\"#\"\"\",&F.F0%\"cGF0,&%\"b GF0*&)F.\"\"$F0)F2F/F0!\"\"7%F2F4F,%'MatrixG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B . Yasskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{Thlapb"lapf"lapfg"lapg"lapl"lapla"laplac" laplacian!"""""lar"""larg"last"layed"lc""""lculu"le"lead """leadi ""leadin" leadingpri "" leadingprinci ""leadingprincipalmi"leadingprincipalminord"leadingprincipalminordet" leadingprincipalminordeterminant""""""left ""leg"len"leng ""lengt"length""" """lenn"lenng"les"""less ""ic"ical"ich"ics"ide"identif"identit"ier""""iffer"iffop"ificat"ified"ify"ig"igf"ii"""iii ""ij"ile"imes"inalg ""inant"inat"includ# """"""""ind"independ+ """"""""""indicat"""""ine" inearalgebra"inert""""""inertlineintscalar"inertlineintvector"inertsurfaceintscalar"inertsurfaceintvector"sforgetg"sg""""""sgf ""sho"short""""show"""sian"sign" significant"similar"""""simpl"simpli ""simplif """""" simplificat "" simplifyvec ""sinC$""""""""""""""""sinc"singC1""""""""""""""""singf+!""" """""""singl"sion ""sis""""sisg"siv """sivg"right"""rincipal"rior"ript ""rist"ritical"rix"rixg"rk"rl"rm"rminant"rming"rms"rng""" " rns"roduct"root"rootof"rootofg"rotat"row """""""rowg/"""" """""""rowgf" rpotential""ibmintelnthyperlinkhelpheadcourinormaltimebulletitemcourierfixedwidthwidthmappfunctionveccalcpackagcallsequenclimitlimdiffveccalcintvaludescriptthescommandexactllikeusualmaplsamenameexceptwhanthinputlistvectormatrixarratheyapplieachomponusecommandyoumustfirstexecutwithcopyrightjeffreyasskinphilipdepartmmathematictexauniversitalsongscalarvalunestarraydescriptdefinvaluedmorevariablarrowformproducefunctionsimilarothertypstructuruseevalfunctevaluatrrayveccalcthesonlyafterperformwithalwayaccesslongcommanexamplfgftgoperatorgarrowgfcurvfgseveraldensitsinlnxgygzgoperatorgsinglngvariablesfieldcoordinattransformatparametricsurfacrtablegmatrixgugvgvectorgcolumngrtableghefafptptgkwagbgcgrtablegcgfgfcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopneutralprecedenc"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~CoordConversion2DCoordConversion3DCurveCurveDotJacobianDeterminant!LeadingPrincipalMinorDeterminants LineIntScalar MakeFunctionMappedFunctions-ScalarPotentialSurfaceSurfaceIntScalar!VectorPotential } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Command:" }{TEXT -1 1 " " }{TEXT 260 24 "VecCalc[evalFunction] - " }{TEXT -1 45 "Evaluate a Func tion using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 9 "Operator :" }{TEXT -1 1 q" " }{TEXT 265 14 "VecCalc[&@] - " }{TEXT -1 45 "Evalua te a Function using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The alias can be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT 256 22 " EF = evalFunction" }}{PARA 0 "" 0 " " {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 92 " evalFunction(fnc,point) fnc &@ point EF(fnc,point ) VecCalc[evalFunction](fnc,point)" }}{PARA 7 "" 0 "" {TEXT -1 230 " CAUTION: There may need to be a space after the operator &@, or else M aple may think subsequent symbols are part of the name of the operator . If the out expression is not a list or Array, you may need to enclos e it in parentheses." }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 258 13 " fnc - " }{TEXT -1 52 "a function to be evaluated, in the form produced by " }{TEXT 263 12 "MakeFunction" }{TEXT -1 1 "." }}{PARA 0 ""r 0 "" {TEXT 259 13 " point - " }{TEXT -1 133 "an expression or expression sequence, li st or Vector of expressions representing the point at which the functi on should be evaluated." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descr iption:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 262 12 "evalFunction " }{TEXT -1 165 " evaluates a function at a point. It is unnecessary f or list- or listlist-valued functions. However, it is essential for V ector-, Matrix- or Array-valued functions." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 270 12 "evalFunction" }{TEXT -1 18 " and \+ the operator " }{TEXT 269 2 "&@" }{TEXT -1 17 " are part of the " } {TEXT 266 7 "VecCalc" }{TEXT -1 78 " package, and so can be used in th ese forms only after performing the command " }{TEXT 267 13 "with(VecC alc)" }{TEXT -1 4 " or " }{TEXT 268 33 "with(VecCalc, evalFunction, `& @`)" }{TEXT -1 55 ". The command can always be accessed in the long f orm " }{TEXT 261 21 "VecCalc[evalFunction]" }{TEXT -1 13 ". The alias " }{TEsXT 264 2 "EF" }{TEXT -1 47 " can be used only after performing \+ the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." } }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {PARA 0 "" 0 "" {TEXT -1 49 "A vector function of one variable (e.g. a curve):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "r:=MakeFunction( t,); evalFunction(r,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"rG-%'RTABLEG6%\")!o:H\"-%'MATRIXG6#7%7#f*6#%\"tG6\"6$%)operatorG%& arrowGF19$F1F1F17#f*F/F1F2F1*$)F5\"\"#\"\"\"F1F1F17#f*F/F1F2F1*$)F5\" \"$F;F1F1F1&%'VectorG6#%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\")+qm6-%'MATRIXG6#7%7#\"\"#7#\"\"%7#\"\")&%'VectorG6#%'colu mnG" }}}{PARA 0 "" 0 "" {TEXT -1 116 "A vector function of several var iables (e.g. a vector field or a coordinate transformation or a parame tric surface):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=MF(,<(u+v)/2, (u-v)/2>); EF(f,2,4), EtF(f,<2,4>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'RTABLEG6%\")Se\"H\"-%'MATRIXG6#7$7#f*6$%\"uG% \"vG6\"6$%)operatorG%&arrowGF2,&*&#\"\"\"\"\"#F99%F9F9*&F8F99$F9F9F2F2 F27#f*F/F2F3F2,&*&F8F9F=F9F9*&#F9F:F9F;F9!\"\"F2F2F2&%'VectorG6#%'colu mnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")gf\"H\"-%'MATRIX G6#7$7#\"\"$7#!\"\"&%'VectorG6#%'columnG-F$6%\")+g\"H\"F'F/" }}}{PARA 0 "" 0 "" {TEXT -1 78 "A Matrix function of several variables evaluate d at a point and along a curve:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=[x,y,z] &-> <,>;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")#*>D8-%'MATRIXG6#7$7%f*6%% \"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3*$)9$\"\"#\"\"\"F3F3F3f*F/F3F4 F3,&F9F;9&F;F3F3F3f*F/F3F4F3,&9%F;*&)F9\"\"$F;)F>F:F;!\"\"F3F3F37%f*F/ F3F4F3F>F3F3F3f*F/F3F4F3FAF3F3F3F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A &@ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\"))=aK\"-%'MATRIXG6#7$7%*$)%#x0G\"\"#\"\"\",&F.F0%#z0uGF 0,&%#y0GF0*&)F.\"\"$F0)F2F/F0!\"\"7%F2F4F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "A&@r&@t;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\")o7E8-%'MATRIXG6#7$7%*$)%\"tG\"\"#\"\"\",&F.F0*$)F.\" \"$F0F0,&F,F0*$)F.\"\"*F0!\"\"7%F2F,F,%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "Fn:= &-> ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FnG-%'RTABLEG6%\")!G;H\"-%'MATRIXG6#7$7#f*6$%\"xG %\"yG6\"6$%)operatorG%&arrowGF2*&)9$\"\"#\"\"\"9%F:F2F2F27#f*F/F2F3F2* &F8F:F;!\"#F2F2F2&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "line:=<4,0>+s*<0,1>; Fn&@line;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lineG,&*&%\"sG\"\"\"-%'RTABLEG6%\")Sk\"H\"-%'MATRIXG 6#7$7#\"\"!7#F(&%'VectorG6#%'columnGF(F(-F*6%\")[bB7-F.6#7$7#\"\"%F1F4 F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?l\"H\"-%'MATRIXG 6#7$7#,$*&\"#;\"\"\"%\"sGF/F/7#,$*&\"\"%F/F0!\"#F/&%'VectorG6#%'column G" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "- Copyright 2003 by Arthur Belmonte and Philip B. Yavsskin\n Department of Mathematics, Texa s A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "MakeFunction" 2 "MakeFunction" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " operator" 2 "operator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "neutral" 2 " neutral" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operators[precedence]" 2 " operators[precedence]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12915680 11667000 12915840 12915960 12916000 13251992 13254188 13261268 12916280 12916440 12235548 12916520 }{RTABLE M7R0 I5RTABLE_SAVE/12915680X*%)anythingG6"6"[gl!#%!!!"$"$f*6#%"tG6"6$%)operatorG%&ar rowGF*9$F*F*F*f*F(F*F+F**$F.""#F*F*F*f*F(F*F+F**$F.""$F*F*F*F* } {RTABLE M7R0 I5RTABLE_SAVE/11667000X*%)anythingG6"6"[gl!#%!!!"$"$""#""%"")6" } {RTABLE M7R0 I5RTABLE_SAVE/12915840X*%)anythingG6"6"[gl!#%!!!"#"#f*6w$%"uG%"vG6"6$%)operatorG %&arrowGF+,&9%#"""""#9$F1F+F+F+f*F(F+F,F+,&F4F1F0#!""F3F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/12915960X*%)anythingG6"6"[gl!#%!!!"#"#""$!""6" } {RTABLE M7R0 I5RTABLE_SAVE/12916000X*%)anythingG6"6"[gl!#%!!!"#"#""$!""6" } {RTABLE M7R0 I5RTABLE_SAVE/13251992X,%)anythingG6"6"[gl!"%!!!#'"#"$f*6%%"xG%"yG%"zG6"6$%)ope ratorG%&arrowGF,*$9$""#F,F,F,f*F(F,F-F,9&F,F,F,f*F(F,F-F,,&F1"""F4F7F,F,F,f*F(F ,F-F,9%F,F,F,f*F(F,F-F,,&F9F7*&F1""$F4F2!""F,F,F,F'F, } {RTABLE M7R0 I5RTABLE_SAVE/13254188X,%)anythingG6"6"[gl!"%!!!#'"#"$*$%#x0G""#%#z0G,&F("""F*F ,%#y0G,&F-F,*&F(""$F*F)!""F'6" } {RTABLE M7R0 I5RTABLE_SAVE/13261268X,%)anythingG6"6"[gl!"%!!!#'"#"$*$%"tG""#*$F(""$,&F("""F* F-F',&F'F-*$F(""*!""F'6" } {RTABLE M7R0 I5RTABLE_SAVE/12916280X*%)anythingG6"6"[gl!#%!!!"#"#f*6$%"xG%"yG6"6$%)operatorG %&arrowGF+*&9$""#9%"""F+F+F+f*F(F+F,F+*&F0F3F2!"#F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/12916440X*%)anythingG6"6"[gl!#%!!!"#"#""!"""6" } {RTABLE M7R0 I5RTABLE_SAVE/12235548X*%)anythingG6"6"[gl!#%!!!"#"#x""%""!6" } {RTABLE M7R0 I5RTABLE_SAVE/12916520X*%)anythingG6"6"[gl!#%!!!"#"#,$%"sG"#;,$*$F(!"#""%6" } I5RTABLE_SAVE/12915960X*%)anythingG6"6"[gl!#%!!!"#"#""$!""6" } {RTABLE M7R0 I5RTABLE_SAVE/12916000X*%)anythingG6"6"[gl!#%!!!"#"#""$!""6" } {RTABLE M7R0 I5RTABLE_SAVE/13251992X,%)anythingG6"6"[gl!"%!!!#'"#"$f*6%%"xG%"yG%"zG6"6$%)ope ratorG%&arrowGF,*$9$""#F,F,F,f*F(F,F-F,9&F,F,F,f*F(F,F-F,,&F1"""F4F7F,F,F,f*F(F ,F-F,9%F,F,F,f*F(F,F-F,,&F9F7*&F1""$F4F2!""F,F,F,F'F, } {RTABLE M7R0 I5RTABLE_SAVE/13254188X,%)anythingG6"6"[gl!"%!!!#'"#"$*$%#x0G""#%#z0G,&F("""F*F ,%#y0G,&F-F,*&F(""$F*F)!""F'6" } {RTABLE M7R0 I5RTABLE_SAVE/13261268X,%)anythingG6"6"[gl!"%!!!#'"#"$*$%"tG""#*$F(""$,&F("""F* F-F',&F'F-*$F(""*!""F'6" } {RTABLE M7R0 I5RTABLE_SAVE/12916280X*%)anythingG6"6"[gl!#%!!!"#"#f*6$%"xG%"yG6"6$%)operatorG %&arrowGF+*&9$""#9%"""F+F+F+f*F(F+F,F+*&F0F3F2!"#F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/12916440X*%)anythingG6"6"[gl!#%!!!"#"#""!"""6" } {RTABLE M7R0 I5RTABLE_SAVE/12235548X*%)anythingG6"6"[gl!#%!!!"#"#x SA,VecCalc" SN,VecCalc" SNL,VecCalc" SPot,VecCalc" ST,VecCalc"ScalarPotential,VecCalc"Sforget,VecCalc" Sis,VecCalc" Siv,VecCalc"SurfaceArea,VecCalc"SurfaceForget,VecCalc"SurfaceIntScalar,VecCalc"SurfaceIntScalarInert,VecCalc"SurfaceIntVector,VecCalc"SurfaceIntVectorInert,VecCalc"SurfaceNormal,VecCalc"SurfaceNormalLength,VecCalc"SurfaceTangents,VecCalc"VCalias,VecCalc" VPot,VecCalc"VectorPotential,VecCalc" c2r,VecCalc" c2s,VecCalc"cyl2rect,VecCalc"cyl2sph,VecCalc""pibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalccurveforgetclearremembtablcurvanalysialiacanusedafterexecutvcaliacommandcforgetcallsequenccurveforgetparametereachthformlistvectorexpresswithparametdescriptveccalcpackagperformfrenetanalsissuchcurvevelocitcurveacceleratusestortheirresultcutsdowncomputotherthesallpartonlywitalwayaccesslongafterexamplveccalsincosrgrtablegvectorgtgsinggfrowgcbgncosgtgfsingfcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocosgf"&ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalcdivergenccalculatvectorfieldarrownotataliacanusedafterexecutionvcaliacommanddivcallsequencvarsdivergencparameterformlistdefinedfunctionvariabloptionalnameindependdescriptactsdefinreturnspecificatunlessunabldeterminhappenconstantbuiltundefindivergencediffercommandlinalgvergdiverginalgpackagexpressrequirspecificativectorcalculuvectorfieldexpressoordinatsystemhepartonlyperformingwithfunctioalwayaccesslongergencperformvcexamplmakefunctfgrtablegxgygzgoperatorgarrowgffbfvectorgrowgcopyrightarthubelmontphilipyasskindepartmmathematictexuniversitalsoncediffopgradicurlectorpotentialvectorpotentialcoo"coor ""coord"coordcon" coordconv" coordconver" coordconvers-""""""" coordconversi"coordi ""coordin ""coordina" coordinat;8"""" """""""""" copyright#"""""""""""""""""""""""""""""""""""corn" correspond"""cos?"""""""""""""""makef"makefun"makefunc "" makefunctK6""" """"""""""""""" makefunctio"mal" mallength"mand""""""many"map"mapfunc"maplg" """"""""""""""""""""""""""""""""""mapp"mappedf" mappedfunct " maps"mat"match"""""math"""""""""""""""""""""""""""""""""""Angle"Mathematics/Packages/VecCalc/AngleAngle+Mathematics/Packages/VecCalc/Commands/AngleAngleConversion,Mathematics/Packages/VecCalc/AngleConversionAngleConversion5Mathematics/Packages/VecCalc/Commands/AngleConversionAngleConversion-Mathematics/Packages/VecCalc/Commands/deg2radAngleConversion-Mathematics/Packages/VecCalc/Commands/rad2degAngleConversion(Mathematics/Packages/VecCalc/Aliases/d2rAngleConversion(Mathematics/Packages/VecCalc/Aliases/r2dCoordConversion2D.Mathematics/Packages/VecCalc/CoordConversion2DCoordConversion2D0Mathematics/Packages/VecCalc/Commands/polar2rectCoordConversion2D0Mathematics/Packages/VecCalc/Commands/rect2polarCoordConversion2D(Mathematics/Packages/VecCalc/Aliases/d2rCoordConversion2D(Mathematics/Packages/VecCalc/Aliases/r2d"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 {CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 15 "VecCalc[Dot] - " }{TEXT -1 40 "Computes the Dot Produ ct of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 9 "Operator:" }{TEXT -1 1 " " }{TEXT 259 14 "VecCalc[&.] - " }{TEXT -1 40 "Computes the Dot Product of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Seque nces:" }{TEXT -1 1 "\n" }{TEXT 257 44 " Dot(u,v) u &. v VecC alc[Dot](u,v)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 11 " u,v - " }{TEXT -1 36 "lists or Vectors of t he same length." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description: " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 260 8 "Dot(u,v)" }{TEXT -1 5 " and " }{TEXT 261 6 "u &. v" }{TEXT -1 46 " calculate the dot produ ct of the two vectors " }{TEXT 262 1 "u" }{TEXT -1 5 " and " }{TEXT 263 1 "v" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 273 12 "VecCalc[Dot ]" }{TEXT -1 18 " is equivalent to " }{TEXT 274 15 "linalg[dotprod]" } {TEXT -1 11 " where the " }{TEXT 23 12 "'orthogonal'" }{TEXT -1 100 " \+ option is always specified except that it work with lists and Vectors \+ instead of lists and vectors." }}{PARA 15 "" 0 "" {TEXT 264 12 "VecCal c[Dot]" }{TEXT -1 18 " is equivalent to " }{TEXT 271 25 "LinearAlgebra [DotProduct]" }{TEXT -1 11 " where the " }{TEXT 272 15 "conjugate=fals e" }{TEXT -1 86 " option is always selected except that it can also wo rk with lists as well as Vectors." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 265 3 "Dot" }{TEXT -1 18 " and the operator " } {TEXT 266 2 "&." }{TEXT -1 17 " are part of the " }{TEXT 267 7 "VecCal c" }{TEXT -1 71 " package, and so can be used by name only after perfo rming the command " }{TEXT 268 13 "with(VecCalc)" }{TEXT -1 4 " or " } {TEXT 269 24 "with(VecCalc, Dot, `&.`)" }{TEXT -1 56 ". The function \+ can always be accessed in the long form " }{TEXT 270 12 "VecCalc[Dot] " }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCal c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "u,v:=<1,2,3>, ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"uG%\"vG6$-%'RTABLEG6%\")/> t5-%'MATRIXG6#7%7#\"\"\"7#\"\"#7#\"\"$&%'VectorG6#%'columnG-F)6%\"(OAH $-F-6#7%7#%\"aG7#%\"bG7#%\"cGF6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Dot(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"*&\" \"#F%%\"bGF%F%*&\"\"$F%%\"cGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &. v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\" \"*&\"\"#F%%\"bGF%F%*&\"\"$F%%\"cGF%F%" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[dotprod ]" 2 "linalg[dotprod]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebr a[DotProduct]" 2 "LinearAlgebra[DotProduct]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Cross" 2 "Cross" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Leng th" 2 "Length" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operator" 2 "operato r" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "neutral" 2 "neutral" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operators[precedence]" 2 "operators[precedenc e]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 10731904 3292236 } {RTABLE M7R0 I5RTABLE_SAVE/10731904X*%)anythingG6"6"[gl!#%!!!"$"$"""""#""$6" } {RTABLE M7R0 I4RTABLE_SAVE/3292236X*%)anythingG6"6"[gl!#%!!!"$"$%"aG%"bG%"cG6" } RA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[dotprod ]" 2 "linalg[dotprod]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebr a[DotProduct]" 2 "LinearAlgebra[DotProduct]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Cross" 2 "Cross" "" }{TEXT -1 2 ", " }{HYPERLNK 17 ""ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemtimesfunctveccalcdotcomputproductvectoroperatorproductcallsequncesveccalcparameterlisthesamelengthdescriptcalculatequivallinalgdotprodorthogonaloptionalwayspecifiexceptworkwithinsteadveccallinearalgebradotproductconjugatfalsselectcanalsoworkwellpartpackagusednameonlyafterperformingcommandaccesslongformexamplugvgrtablegmatrixgvectorgcolumngoahagbgcgfbgfcopyrightarthurbelmontphilipasskindepartmmathematictexauniversitlinearalgebrcroslengthoperatoneutralprecedenccg"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 18 "VecCalc[Length] - " }{TEXT -1 35 "Calculate the Lengt h of a Vector " }}{PARA 0 "" 0 "" {TEXT 26 20 "Calling Sequences: \n " }{TEXT 256 35 " Length(v) VecCalc[Length](v)" }}{PARA 0 "" 0 " " {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 9 " v - " }{TEXT -1 18 "a list or Vector. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT 259 9 "Length(v)" }{TEXT -1 147 " calculates the length or ma gnitude or 2-norm of the vector v defined as the square root of the su m of the squares of the components of the vector." }}{PARA 15 "" 0 "" {TEXT 260 9 "Length(v)" }{TEXT -1 14 " differs from " }{TEXT 261 17 "l inalg[norm](v,2)" }{TEXT -1 311 " in that norm(v,2) computes the squar e root of the sum of the squares of the absolute value of the componen ts of the vector. These differ if the components are complex so that \+ a square is not the same as the square of the absolute value. The abs olute values also prevent the simplification of trig identities." }} {PARA 15 "" 0 "" {TEXT 262 9 "Length(v)" }{TEXT -1 18 " is equivalent \+ to " }{TEXT 263 46 "LinearAlgebra[VectorNorm](v,2,conjugate=false)" } {TEXT -1 60 " except that it can also work with lists as well as Vecto rs." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 264 6 "Leng th" }{TEXT -1 16 " is part of the " }{TEXT 265 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 266 6 "Length" } {TEXT -1 35 " only after performing the command " }{TEXT 267 13 "with( VecCalc)" }{TEXT -1 4 " or " }{TEXT 268 21 "with(VecCalc, Length)" } {TEXT -1 56 ". The function can always be accessed in the long form \+ " }{TEXT 269 15 "VecCalc[Length]" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "v:=[a,b,c];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with (VecCalc): Length(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(*$)%\"aG \"\"#\"\"\"F)*$)%\"bGF(F)F)*$)%\"cGF(F)F)#F)F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "linalg[norm](v,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(*$)-%$absG6#%\"aG\"\"#\"\"\"F,*$)-F(6#%\"bGF+F,F,*$ )-F(6#%\"cGF+F,F,#F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u :=;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG 6%\")[iP8-%'MATRIXG6#7$7#-%$sinG6#%\"tG7#-%$cosGF0&%'VectorG6#%'column G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Length(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "linalg[norm](u,2); simplify(%); # t may be complex" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)-%$absG6#-%$sinG6#%\"tG\"\"#\" \"\"F/*$)-F(6#-%$cosGF,F.F/F/#F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$,&*$)-%$absG6#-%$sinG6#%\"tG\"\"#\"\"\"F/*$)-F(6#-%$cosGF,F.F/F/#F/F ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(%) assuming r eal;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "LinearAlgebra[Norm](u,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)-%$absG6#-%$sinG6#%\"tG\"\"#\"\"\"F/*$)-F(6#-%$c osGF,F.F/F/#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "LinearA lgebra[Norm](u,2,conjugate=false); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)-%$sinG6#%\"tG\"\"#\"\"\"F,*$)-%$cosGF)F+F,F,#F, F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B . Yasskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 " VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[norm]" 2 "linalg[norm]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[Norm ]" 2 "LinearAlgebra[Norm]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Dot" 2 " Dot" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13376248 }{RTABLE M7R0 I5RTABLE_SAVE/13376248X*%)anythingG6"6"[gl!#%!!!"#"#-%$sinG6#%"tG-%$cosGF)6" } A 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)-%$absG6#-%$sinG6#%\"tG\"\"#\"\"\"F/*$)-F(6#-%$c osGF,F.F/F/#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "LinearA lgebra[Norm](u,2,conjugate=false); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)-%$sinG6#%\"tG\"\"#\"\"\"F,*$)-%$cosGF)F+F,F,#F, F+" }}{PARA val"valf"valuK?""""""""""""""""""valuat"vandermondematrix"var$"""""" " vari"varia ""variab"variablS_""""""""""""""""""""vars/6"""""""""""vativ"vc"""""vca"""vcalia{s""""""""""""""""""""""""""""""ve7"""""""""""""vec# """"""""vecc' """""""""vecca"""""^ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaloutputbulletitemfunctveccalcgradicalculatgradientarrownotataliacanusedafterexecutvcaliacommandgradcallsequencvarsouttypparameterscalarformdefinvariablesoptionallistvectornamesindependvariablyperowcolumdescriptthfirstpartialderivatactsunctreturnfunctionspecificatunlesunabldeterminhappenconstantbuiltundefinparametspecificonverthavetypeotherwisoutputglobaloutputvectortypoutputvectortypdefaultsetdifferfrolinalgpackagexpressexpressionrequirspecificatvectorcalculuvectorcalculureturnvectorfieldcoordinatsystempartackagonlyperformwithalwayaccesslongexamplfgfxgygzgoperatorgarrowgfdelfdelfgrtablegvectorgoperatorgfdfrowgevalfunctgtxgfmakefunctexpggfexpgdelgcolumndelggoppmatrixgagbgngefqpqgfpgfvectorgcolumngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitsealsolineare"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 17 "VecCalc[Angle] - " }{TEXT -1 39 "Computes the Angle b etween Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences: " }{TEXT -1 1 "\n" }{TEXT 257 37 " Angle(u,v) VecCalc[Angle](u,v )" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 11 " u,v - " }{TEXT -1 36 "lists or Vectors of the same \+ length." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 279 5 "Angle" }{TEXT 260 5 "(u,v)" } {TEXT -1 42 " calculates the angle between two vectors " }{TEXT 262 1 "u" }{TEXT -1 5 " and " }{TEXT 263 1 "v" }{TEXT -1 1 "." }}{PARA 15 " " 0 "" {TEXT 273 8 "VecCalc[" }{TEXT 280 5 "Angle" }{TEXT 281 1 "]" } {TEXT -1 18 " is equivalent to " }{TEXT 274 13 "linalg[angle]" }{TEXT -1 73 " except that it work with lists and Vectors instead of lists an d vectors." }}{PARA 15 "" 0 "" {TEXT 264 8 "VecCalc[" }{TEXT 282 5 "An gle" }{TEXT 283 1 "]" }{TEXT -1 18 " is equivalent to " }{TEXT 271 20 "LinearAlgebra[Vector" }{TEXT 285 5 "Angle" }{TEXT 286 1 "]" }{TEXT -1 11 " where the " }{TEXT 272 15 "conjugate=false" }{TEXT -1 86 " opt ion is always selected except that it can also work with lists as well as Vectors." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 284 5 "Angle" }{TEXT -1 16 " is part of the " }{TEXT 267 7 "VecCalc" } {TEXT -1 71 " package, and so can be used by name only after performin g the command " }{TEXT 268 13 "with(VecCalc)" }{TEXT -1 4 " or " } {TEXT 269 14 "with(VecCalc, " }{TEXT 287 5 "Angle" }{TEXT 288 1 ")" } {TEXT -1 56 ". The function can always be accessed in the long form \+ " }{TEXT 270 8 "VecCalc[" }{TEXT 289 5 "Angle" }{TEXT 290 1 "]" } {TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCal c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "u,v:=<1,1,0>, <0,1,1 >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"uG%\"vG6$-%'RTABLEG6%\")![ 3*=-%'MATRIXG6#7%7#\"\"\"F07#\"\"!&%'VectorG6#%'columnG-F)6%\")3Y'*=-F -6#7%F2F0F0F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Angle(u,v) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%#PiG\"\"\"F(" }}} {EXCHG }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematic s, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "linalg[angle]" 2 "linalg[angle]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[VectorAngle]" 2 "LinearAlgebra[VectorAngl e]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Dot" 2 "Dot" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Length" 2 "Length" "" }{TEXT -1 1 "." }}}}{MARK "0 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 18908480 18964608 }{RTABLE M7R0 I5RTABLE_SAVE/18908480X*%)anythingG6"6"[gl!#%!!!"$"$"""F'""!F& } {RTABLE M7R0 I5RTABLE_SAVE/18964608X*%)anythingG6"6"[gl!#%!!!"$"$""!"""F(F& } F07#\"\"!&%'VectorG6#%'columnG-F)6%\")3Y'*=-F -6#7%F2F0F0F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Angle(u,v) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%#PiG\"\"\"F(" }}} {EXCHG }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematic s, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "linalg[angle]" 2 "linalg[angle]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[VectorAngle]" 2 "Li"hibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfixewidthfunctionveccalcsurfaceintvectorinertdisplaysurfacintegralvectorfieldsurfaceintvectorcomputesbothcommandcandisplaintermediatstepaliasusedafterxecutvcaliavcsivcallsequencvarrngsurfaceintvectorinertsurfaceintvectorinertarsurfaceintvectorparameterformlistfunctionsvariablarrownotatparametricdefinintegratcurvrangintegratoveoptionalparametindicatdisplaydescriptwherefirstargumsecondthirdfourthargumentintegrationtheirevaluatusingvaluevalfcalculatithoutincludcalculateswhilallreturninertmultiplreturnsyoudoneedquotaroundassignappeareithorderwhatevbesthowevnameintegrationmustmatchdefinittheydifferentialsimilarcommandvectorcalculufluxpackagwithdomainchosenusesexpressnsteadcannotnorshorequirspecificatcoordinatsystemthespartusedonlyperformveccalcalwayaccesslongxamplmakefunc"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaltimesbulletitemfrenetanalysicurvusingveccalcpackagcallsequenccommandaliaparameterlistbelowlowformvectorexpresswithparametsimplifsuchdescriptthesdesignperformeachshortmostworkanydimenshowevcurvebinormalcurvetorsonlycorrespondactioncurvevelocitcvcalculatvelocitcurveacceleratcaacceleratcurvejerkcjjerkcurvespecsspeedcurvearclengthclarclengthtangctunitangentcurvenormalcnprincipalcbbinormalcurvecurvaturckcurvaturcurvetorstorsioncurvetangentialacceleratcattangentialacceleraticurvenormalacceleratcurveforgetcforgetclearremembtablabovusenameyoumustfirstexecutaliasmustvcaliaexceptwillattemptcurvemayplottplotparametricargumdimensionspacecurvusesupcomputatafterfinishingavoidcluttermemorthidoneexamplcossinrgrtablegvectorgtgcosgsingfrowgpigvingfswingtgfcosgfvectoderivatcosgvalulgfagbgoCurveNormalAccelerationOutputVectorType VecCalc,&.VecCalc,CurveCurvatureVecCalc,Length VecCalc,VPotVecCalc,rect2polar evalFunction"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MakeFunction'Mathematics/Packages/VecCalc/Aliases/MFMappedFunctions,Mathematics/Packages/VecCalc/MappedFunctionsMappedFunctions+Mathematics/Packages/VecCalc/Commands/LimitMappedFunctions+Mathematics/Packages/VecCalc/Commands/limitMappedFunctions*Mathematics/Packages/VecCalc/Commands/DiffMappedFunctions*Mathematics/Packages/VecCalc/Commands/diffMappedFunctions)Mathematics/Packages/VecCalc/Commands/IntMappedFunctions)Mathematics/Packages/VecCalc/Commands/intackages/VecCalc/Commands/&-> }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 17 "VecCalc[Cross] - " }{TEXT -1 45 "Calculates the Cross Product of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 10 "Operator: " }{TEXT 259 17 "VecCalc[&x] - " }{TEXT -1 44 "Calculates the Cross P roduct of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequenc es:" }{TEXT -1 1 "\n" }{TEXT 256 68 " Cross(u, v, outtype) u &x \+ v VecCalc[Cross](u, v, outtype)" }}{PARA 7 "" 0 "" {TEXT -1 139 "C AUTION: There must be a space after the operator &x, or else Maple wil l think the subsequent letters are part of the name of the operator." }{TEXT 18 0 "" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 257 13 " u, v - " }{TEXT -1 43 "lists or Vectors, e ach with three elements." }}{PARA 0 "" 0 "" {TEXT 270 13 " outtype - " }{TEXT 271 24 "(optional) output type: " }{TEXT 272 51 "'list', 'Ve ctor', 'Vector[row]' or 'Vector[column]'" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 260 10 "cross(u,v)" }{TEXT -1 5 " and " }{TEXT 261 6 "u &x v" }{TEXT -1 124 " calculate the cross product of two 3-dimensional lists or Vec tors and returns the answer as a 3-dimensional list or Vector." }} {PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 273 7 "outtype" }{TEXT -1 84 " parameter is specified, then the output is converted to have t he type specified by " }{TEXT 274 7 "outtype" }{TEXT -1 59 ". Otherwis e the output type matches the type of the vector " }{TEXT 275 1 "v" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 276 14 "VecCalc[Cross]" } {TEXT -1 18 " is equivalent to " }{TEXT 277 17 "linalg[crossprod]" } {TEXT -1 75 " except that it works with lists and Vectors instead of l ists and vectors. " }}{PARA 15 "" 0 "" {TEXT 262 14 "VecCalc[Cross]" } {TEXT -1 18 " is equivalent to " }{TEXT 263 27 "LinearAlgebra[CrossPro duct]" }{TEXT -1 61 " except that it can also work with lists as well \+ as Vectors. " }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 264 5 "Cross" }{TEXT -1 18 " and the operator " }{TEXT 265 2 "&x" } {TEXT -1 17 " are part of the " }{TEXT 266 7 "VecCalc" }{TEXT -1 71 " \+ package, and so can be used by name only after performing the command \+ " }{TEXT 267 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 268 26 "with (VecCalc, Cross, `&x`)" }{TEXT -1 56 ". The function can always be ac cessed in the long form " }{TEXT 269 14 "VecCalc[Cross]" }{TEXT -1 1 " ." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u:=[1,2,3]; v:=[a,b,c];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Cross(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*& \"\"#\"\"\"%\"cGF'F'*&\"\"$F'%\"bGF'!\"\",&*&F*F'%\"aGF'F'F(F,,&F+F'*& F&F'F/F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Cross(u,v,Vec tor[column]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")gR\"H \"-%'MATRIXG6#7%7#,&*&\"\"#\"\"\"%\"cGF/F/*&\"\"$F/%\"bGF/!\"\"7#,&*&F 2F/%\"aGF/F/F0F47#,&F3F/*&F.F/F8F/F4&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &x v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&\"\"#\"\"\"%\"cGF'F'*&\"\"$F'%\"bGF'!\"\",&*&F*F' %\"aGF'F'F(F,,&F+F'*&F&F'F/F'F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "V ecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[crossprod]" 2 "linal g[crossprod]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[CrossPr oduct]" 2 "LinearAlgebra[CrossProduct]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Dot" 2 "Dot" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operator " 2 "operator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "neutral" 2 "neutral " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operators[precedence]" 2 "operato rs[precedence]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12913960 }{RTABLE M7R0 I5RTABLE_SAVE/12913960X*%)anythingG6"6"[gl!#%!!!"$"$,&%"cG""#%"bG!"$,&%"aG""$F( !"",&F*"""F-!"#6" } 6#7%,&*&\"\"#\"\"\"%\"cGF'F'*&\"\"$F'%\"bGF'!\"\",&*&F*F' %\"aGF'F'F(F,,&F+F'*&F&F'F/F'F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "V ecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[crossprod]" 2 "linal g[crossprod]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[CrossPr oduct]"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalccurlcalculatdimensionalvectorfieldarrownotatcallsequencvarsouttypouttypeparameterdimenionalformlistdefinfunctionvariabloptionalnameusedindependrowcolumndescriptactsreturnfinedpecificatunlesunabldeterminecanhappenfunctionsconstantbuiltundefinparametspecificonverthaveheotherwimatchdiffercommandlinalgpackagexpressionsexpressrequirspecificationvectorcalculuvectorcalculuvectorfieldexpressionoordinatsystempartthonlyafterperformwithalwayccesslongexamplvcaliamakefunctfgxgygzgoperatorgarrowgfcfcfgbgagfefrtablegssmatrifcffdfffffefbfvectorgolumngsumatrixgcgcolumngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopgradidivergencscalarpotentialvectorpotentialrtablsaveanythinggglag"<ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriertimesnormalutputbulletitemmultivariablmaxminproblemusingveccalcpackagfunctiongradiveccalccalculatstudequatgradsetequalzerouplagrangmultipliequationsolvexactriticalpointfsolvapproximatdecimalcriticalallvaluevaluatsolutioncontarootofhessianveccalcleadingprincipalminordeterminantleadingprincipalminordterminantleadingprincipalminordeterminantapplsecondderivattestsubsconvertsolutiintolistcoordinatmakefunctmakearrowdefinfunctveccalcmakeevalfunctvaluataliathesaliascanusedafterexecutvcaliacommandhesslpmdmfdescriptiondesignwithmultidimensionalbelowexamplbothunconstrainconstrainuseleadingprincipalminordeterminantmakefunctionoperatoryoumustfirstwitnamecommandfindallclassifeachlocalmaximumminimumsaddlexpfgfxgygoperatorgarrowgfexpgsolutcomputsoldelfdelfgrtablegovvectorgfcffbffnfcfgfrowgeqseqsgxgfU evalfunctio"evalg """""evalu"evalua"evaluat+""""""""""ex ""exac"exact""""exactl"exam"examp ""exampl%"""""""""""""""""""""""""""""""exc"except """""""execu"executg"""""""""""""""""""""""""exist"""exp """""expg0"#""""{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 23 "VecCalc[Determinant] - " }{TEXT -1 37 "Calculate the \+ Determinant of a Matrix" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The alias can be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 265 22 " Det = Determinant" }}{PARA 0 "" 0 "" {TEXT 26 20 "C alling Sequences: \n" }{TEXT 256 52 " Determinant(M) Det(M) VecC alc[Determinant](M)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 9 " M - " }{TEXT -1 43 "a square list of lists or a square Matrix. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT 266 14 "Determinant(M)" }{TEXT -1 47 " calculates the determinan t os a square matrix." }}{PARA 15 "" 0 "" {TEXT 267 11 "Determinant" } {TEXT 259 3 "(M)" }{TEXT -1 24 " is exactly the same as " }{TEXT 260 14 "LinearAlgebra[" }{TEXT 268 15 "Determinant](M)" }{TEXT -1 69 " exc ept that it can also act on a lists of lists as well as a Matrix." }} {PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 269 11 "Determinan t" }{TEXT -1 16 " is part of the " }{TEXT 261 7 "VecCalc" }{TEXT -1 42 " package, and so can be used in the form " }{TEXT 270 11 "Determi nant" }{TEXT -1 35 " only after performing the command " }{TEXT 262 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 263 14 "with(VecCalc, " } {TEXT 271 11 "Determinant" }{TEXT 272 1 ")" }{TEXT -1 56 ". The funct ion can always be accessed in the long form " }{TEXT 264 8 "VecCalc[" }{TEXT 273 11 "Determinant" }{TEXT 274 1 "]" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "M:=LinearAlgebra[VandermondeMatrix] ();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\")!=; K\"-%'MATRIXG6#7%7%\"\"\"%\"aG*$)F/\"\"#F.7%F.%\"bG*$)F4F2F.7%F.%\"cG* $)F8F2F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Determ inant(M); factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&%\"bG\"\" \")%\"cG\"\"#F&F&*&)F%F)F&F(F&!\"\"*&F(F&)%\"aGF)F&F&*&F/F&F'F&F,*&F/F &F+F&F&*&F%F&F.F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"cG!\" \"%\"bG\"\"\"F),&%\"aGF)F&F'F),&F+F)F(F'F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "N:=[[a,b,c],[d,e,f],[g,h,i]];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"NG7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"h G%\"iG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(%\"aG\"\"\"%\"eGF&%\"iGF&F&*(F%F &%\"fGF&%\"hGF&!\"\"*(%\"dGF&F+F&%\"cGF&F&*(F.F&%\"bGF&F(F&F,*(%\"gGF& F1F&F*F&F&*(F3F&F'F&F/F&F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n D epartment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecC alc" "" }{TEXT -1 1 "," }{TEXT -1 1 " " }{HYPERLNK 17 "LinearAlgebra[D eterminant]" 2 "LinearAlgebra[Determinant]" "" }{TEXT -1 1 "." }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13216180 }{RTABLE M7R0 I5RTABLE_SAVE/13216180X,%)anythingG6"6"[gl!"%!!!#*"$"$"""F'F'%"aG%"bG%"cG*$F("" #*$F)F,*$F*F,6" } " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "N:=[[a,b,c],[d,e,f],[g,h,i]];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"NG7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"h G%\"iG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(%\"aG\"\"\"%\"eGF&%\"iGF&F&*(F%F &%\"fGF&%\"hGF&!\"\"*(%\"dGF&F+F&%\"cGF&F&*(F.F&%\"bGF&F(F&F,*(%\"gGF& F1F&F*F&F&*(F3F&F'F&F/F&F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n D epartment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecC alc" "" }{TEXT -1 1 "," }{TEXT -1 1 " " }{HYPERLNK 17 "LinearAlgebra[D eterminant]" 2 "LinearAlgebra[Determinant]" "" }{TEXT -"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaloutputbulletitemfunctveccalchessiancalculathessianarrownotatliasaliacanusedafterexecutvcaliacommandhesscallsequencvarsouttyparameterscalarformdefinvariabloptionallistvectornameindependvariabloutputypelistlistmatrixdescriptsecondpartialderivatactsreturnfunctionspecificatvariablunlesunabldeterminhappenconstantbuiltundefinparametspecifiedconverthavespecifiotherwisspecifiglobaloutputmatrixtypoutputmatrixtypdefaultsetdifferlinalgpackagwhichexpressrequirevectorcalculumatrirequirspecificatipartveccalconlyperformwithwitalwayaccesedlongafterexamplveccalfgfxgygzgoperatorgarrowgfhfhfgrtablegmatrixgfeffaffdffpfffffkffgfqfftfevalfunctbgcgagffbffcmakefunctexpggfexpghghggagqgfpgfopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoveccalclinearalgebrainearalgebradiffotturn"typ"type'!"""""""""udf"ued"uenc"uf"ug+""""""""""ugf"uire"ultipl"umber"un ""unabl' """""""""unapp" unavailabl" unconstrain"unct"""unction ""undefin' """"""""" undefinedg ""uni"""""unit" universit"""""""""""""""""""""""""""""""unles' """"""""" unnecessar"up"""updat""{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 259 45 "VecCalc[LeadingPrincipalMinorDeterminants] - " } {TEXT -1 63 "Calculates the Leading Principal Minor Determinants of a \+ Matrix" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The ali as can be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCa lias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 257 45 " LP MD = LeadingPrincipalMinorDeterminants" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 258 97 " LeadingPrincipalMinorDeterminants(M) LPMD(M) VecCalc[Leadin gPrincipalMinorDeterminants](M)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parame ters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 9 " M - " } {TEXT -1 49 "a square list of lists or Matrix of expressions. " }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 35 "The leading principal minors of an " } {TEXT 260 5 "n x n" }{TEXT -1 23 " square matrix are the " }{TEXT 261 5 "k x k" }{TEXT -1 47 " square submatrices in the top left corner for " }{TEXT 262 9 "k = 1...n" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 167 "The leading_principal_minor_determinants command computes and \+ displays the determinants of the leading principal minors and returns \+ the sequence of these determinants." }}{PARA 15 "" 0 "" {TEXT -1 112 " In linear algebra: A matrix is positive definite if its leading princ ipal minor determinants are all positive. " }}{PARA 15 "" 0 "" {TEXT -1 297 "In multivariable calculus: A critical point p of a function f \+ is a local minimum if the leading principal minor determinants of the \+ Hessian matrix of f at p are all positive. It is a local maximum if t he leading principal minor determinants of the Hessian alternate signs beginning with negative." }}{PARA 15 "" 0 "" {TEXT -1 28 "This comman d is part of the " }{TEXT 263 7 "VecCalc" }{TEXT -1 41 " package, and \+ so can be used in the form " }{TEXT 264 33 "LeadingPrincipalMinorDeter minants" }{TEXT -1 35 " only after performing the command " }{TEXT 265 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 266 14 "with(VecCalc, " }{TEXT 267 34 "LeadingPrincipalMinorDeterminants)" }{TEXT -1 55 ". \+ The command can always be accessed in the long form " }{TEXT 268 42 " VecCalc[LeadingPrincipalMinorDeterminants]" }{TEXT -1 13 ". The alias " }{TEXT 269 4 "LPMD" }{TEXT -1 47 " can be used only after performin g the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "M:=<<4|2|0|-1>, <-3|5|1|0>, \+ <0|2|-3|1>, <2|-4|1|3>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'R TABLEG6%\")w-e8-%'MATRIXG6#7&7&\"\"%\"\"#\"\"!!\"\"7&!\"$\"\"&\"\"\"F0 7&F0F/F3F57&F/!\"%F5\"\"$%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D1,D2,D3,D4:=LPMD(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6&%#D1G%#D2G%#D3G%#D4G6&\"\"%\"#E!#')!$3$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M2:=LinearAlgebra:-SubMatrix(M,1..2,1..2); De t(M2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'RTABLEG6%\"(35@$-%' MATRIXG6#7$7$\"\"%\"\"#7$!\"$\"\"&%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M3:= LinearAlgebra:-SubMatrix(M,1..3,1..3); Det(M3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'RTABLEG6%\");[e8-%'MATRIXG6#7%7%\"\"%\"\"#\"\" !7%!\"$\"\"&\"\"\"7%F0F/F2%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #!#')" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Det(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$3$" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "V ecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Determinant" 2 "Determinant " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[SubMatrix]" 2 "Line arAlgebra[SubMatrix]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin" 2 "MultiMaxMin" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Hessian" 2 "Hessian " "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13580276 3211008 13584816 }{RTABLE M7R0 I5RTABLE_SAVE/13580276X,%)anythingG6"6"[gl!"%!!!#1"%"%""%!"$""!""#F*""&F*!"%F)" ""F(F-!""F)F-""$6" } {RTABLE M7R0 I4RTABLE_SAVE/3211008X,%)anythingG6"6"[gl!"%!!!#%"#"#""%!"$""#""&6" } {RTABLE M7R0 I5RTABLE_SAVE/13584816X,%)anythingG6"6"[gl!"%!!!#*"$"$""%!"$""!""#""&F*F)"""F(6 " } RA 11 "" 1 "" {XPPMATH 20 "6#!$3$" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "V ecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Determinant" 2 "Determinant " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[SubMatrix]" 2 "Line arAlgebra[SubMatrix]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin" 2 "MultiMaxMin" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Hessian" 2 "Hessian " "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13580276 3211008 13584816 }{RTABLE M7R0 I5RTABLE_SAVE/13580276X,%)anythingG6"6"[gl!"%!!!#1"%"%""%!"$""!""#F*""&F*!"%F)" ""F(F-!""F)F-""$6" } {RTABLE M7R0 I4"Hibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalcleadingprincipalminordeterminantcalculatleadprincipalminordeterminantmatrixaliaalicanusedafterexecutvcaliavcaliascommandlpmdcallsequenclpmdleadingprincipalminordeterminantparamterssquarlistexpressdescriptsubmatrictopleftcorncomputdisplayreturntheslinearalgebrapositdefinitprincipalallmultivariablcalculucriticalpointlocalminimumhessianatmaximumhealternatsignbeginnwithnegatcommanpartpackagformleadingprincipalminordetminantonlyperformalwayaccesslongperforminexamplmgtablegmatrixglinearalgebrasubmatrixdertablegdetcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoeccalclinearalgebramultimaxminrtablsaveanythingggl"-{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 CoordConversion2D7Mathematics/Packages/VecCalc/Commands/CoordConversion2DCoordConversion3D.Mathematics/Packages/VecCalc/CoordConversion3DCoordConversion3D7Mathematics/Packages/VecCalc/Commands/CoordConversion3DCoordConversion3D.Mathematics/Packages/VecCalc/Commands/cyl2rectCoordConversion3D.Mathematics/Packages/VecCalc/Commands/rect2cylCoordConversion3D.Mathematics/Packages/VecCalc/Commands/sph2rectCoordConversion3D.Mathematics/Packages/VecCalc/Commands/rect2sphCoordConversion3D-Mathematics/Packages/VecCalc/Commands/cyl2sphCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/d2rCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/r2dCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/d2rCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/r2dCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/d2rCoordConversion3D(Mathematics/Packages/VecCalc/Aliases/r2d)Mathematics/Packages/VecCalc/Aliases/DivDivergence(Mathematics/Packages/VecCalc/Aliases/EFevalFunction*Mathematics/Packages/VecCalc/Aliases/Grad Gradient*Mathematics/Packages/VecCalc/Aliases/Hess Hessian*Mathematics/Packages/VecCalc/Aliases/JDetJacobianDeterminant)Mathematics/Packages/VecCalc/Aliases/JacJacobianMatrix*Mathematics/Packages/VecCalc/Aliases/LPMD&LeadingPrincipalMinorDeterminants)Mathematics/Packages/VecCalc/Aliases/LapLaplacian)Mathematics/Packages/VecCalc/Aliases/LisLineIntScalar)Mathematics/Packages/VecCalc/Aliases/LivLineIntVector(Mathematics/Packages/VecCalc/Aliases/MFMakeFunction+Mathematics/Packages/VecCalc/Aliases/MuintMultipleint(Mathematics/Packages/VecCalc/Aliases/SA Surface} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width " 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 29 " \+ Angle Conversions using the " }{TEXT 267 7 "VecCalc" }{TEXT -1 8 " Pac kage" }}{PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" } {TEXT 256 22 " VecCalc[deg2rad] - " }{TEXT -1 40 "Converts Angles fr om Degrees to Radians " }}{PARA 0 "" 0 "" {TEXT 257 22 " VecCalc[rad 2deg] - " }{TEXT -1 39 "Converts Angles from Radians to Degrees" }} {PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliases can be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 18 " d2r = deg2rad" }}{PARA 259 "" 0 "" {TEXT -1 18 " r2d = rad2deg" }} {PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 60 " deg2rad(theta) d2r(theta) Ve cCalc[deg2rad](theta)" }}{PARA 256 "" 0 "" {TEXT -1 60 " rad2deg(the ta) r2d(theta) VecCalc[rad2deg](theta)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 258 13 " theta \+ - " }{TEXT -1 55 "a number, variable or expression representing an ang le " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 259 7 "deg2rad" }{TEXT -1 68 " converts angles measured in degrees to angles measured in radians. " }}{PARA 15 "" 0 "" {TEXT 260 7 "rad2deg" }{TEXT -1 68 " converts angles measur ed in radians to angles measured in degrees. " }}{PARA 15 "" 0 "" {TEXT -1 59 "If theta contains any floating point decimal numbers, the n " }{TEXT 261 7 "deg2rad" }{TEXT -1 5 " and " }{TEXT 262 7 "rad2deg" }{TEXT -1 88 " return decimal answers. Otherwise, they return exact n umbers or symbolic expressions. " }}{PARA 15 "" 0 "" {TEXT -1 32 "Thes e functions are part of the " }{TEXT 263 7 "VecCalc" }{TEXT -1 71 " pa ckage, and so can be used by name only after performing the command " }{TEXT 264 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 265 22 "with(V ecCalc,function)" }{TEXT -1 58 ". The functions can always be accesse d in the long forms " }{TEXT 266 17 "VecCalc[function]" }{TEXT -1 61 " . The aliases can be used only after performing the command " } {HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "deg2rad(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* (\"$!=!\"\"%\"aG\"\"\"%#PiGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "rad2deg(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$!=\"\"\" %\"aGF&%#PiG!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2 rad(45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%#PiG\"\"\"F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rad2deg(Pi/6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "deg2rad(45.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ N;)R&y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rad2deg(1.); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]zdHd!\")" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Phi lip B. Yasskin\n Department of Mathematics, Texas A&M University \+ " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "C oordConversion2D" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion3D " 2 "CoordConversion3D" "" }{TEXT 26 1 "." }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$!=\"\"\" %\"aGF&%#PiG!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2 rad(45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%#PiG\"\"\"F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rad2deg(Pi/6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "deg2rad(45.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ N;)R&y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rad2deg(1.); \+ " }}{PARA 11 "" 1 "" "uibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfixedwidthanglconversusingveccalcpackagefunctiondegradconvertfromdegreradianaliascanusedafterexecutvcaliacommandcallsequencthetaveccalctaparameternumbvariablexpressrepresentangledescriptmeasuredcontainanyfloatpointdecimalnumberreturnanswerotherwistheyexactumbersymbolicthespartpackagnameonlyperformwitheccalcfunctalwayaccesslongformexamplagpigfagfpigpizdhdcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconversoordconvers"@{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 "%{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1  }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 48 " \+ 2 Dimensional Coordinate Conversions using the " }{TEXT 264 7 "VecCalc " }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT 26 11 "Functions: " } {TEXT -1 1 "\n" }{TEXT 256 25 " VecCalc[polar2rect] - " }{TEXT -1 46 "Converts Coordinates from Polar to Rectangular" }}{PARA 0 "" 0 "" {TEXT 257 25 " VecCalc[rect2polar] - " }{TEXT -1 46 "Converts Coordi nates from Rectangular to Polar" }}{PARA 0 "" 0 "" {TEXT 26 8 "Aliases :" }{TEXT -1 50 " - The aliases can be used after execution of the " } {HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 257 "" 0 "" {TEXT -1 21 " p2r = polar2rect" }}{PARA 257 "" 0 "" {TEXT -1 21 " r2p = rect2polar" }}{PARA 0 "" 0 "" {TEXT 26 18 "Cal ling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 74 " p olar2rect([r,theta]) p2r([r,theta]) VecCalc[polar2rect]([r,theta]) " }}{PARA 256 "" 0 "" {TEXT -1 216 " polar2rect() p2r() VecCalc[polar2rect]()\n rect2polar([x,y]) r 2p([x,y]) VecCalc[rect2polar]([x,y])\n rect2polar() \+ r2p() VecCalc[rect2polar]()" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 258 41 " [x,y] or or - " }{TEXT -1 25 "rectangular coordinates \n" } {TEXT 259 17 " x - " }{TEXT -1 50 "the horizontal coordina te, positive on the right \n" }{TEXT 260 17 " y - " } {TEXT -1 42 "the vertical coordinate, positive upward \n" }{TEXT 261 41 " [r,theta] or or - " }{TEXT -1 19 "polar coo rdinates \n" }{TEXT 262 17 " r - " }{TEXT -1 37 "the radia l distance from the origin \n" }{TEXT 263 17 " theta - " } {TEXT -1 72 "the angle measured in radians counterclockwise from the p ositive x-axis " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description: " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 269 10 "polar2rect" }{TEXT -1 97 " converts polar coordinates to rectangular coordinates using th e formulas:\n " }{XPPEDIT 19 1 "x = r cos(theta); " "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 12 " \+ " }{XPPEDIT 19 1 "y = r sin(theta) ;" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6# %&thetaGF'" }{TEXT -1 67 " \nThere is no restriction on the values o f the polar coordinates." }}{PARA 15 "" 0 "" {TEXT 270 10 "rect2polar " }{TEXT -1 128 " converts rectangular coordinates to polar coordinate s using the formulas:\n \+ " }{XPPEDIT 19 1 "theta = arctan(y/x);" "6#/%&thetaG-%'arctanG6# *&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 42 " in quadrants I and IV \n \+ " }{XPPEDIT 19 1 "r = sqrt(x^2 + y^2) ;" "6#/%\"rG-%%sqrtG 6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT -1 13 " " } {XPPEDIT 19 1 "theta = arctan(y/x) +Pi;" "6#/%&thetaG,&-%'arctanG6#*&% \"yG\"\"\"%\"xG!\"\"F+%#PiGF+" }{TEXT -1 71 " in quadrant II \n \+ " }{XPPEDIT 19 1 "theta =arctan(y/x )-Pi;" "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+ %#PiGF-" }{TEXT -1 80 " in quadrant III \nThe resulting polar coordin ates are restricted to the ranges " }{XPPEDIT 19 1 "r >= 0" "6#1\"\"!% \"rG" }{TEXT -1 5 " and " }{XPPEDIT 19 1 " -Pi < theta " "6#2,$%#PiG! \"\"%&thetaG" }{XPPEDIT 19 1 "``< Pi" "6#2%!G%#PiG" }{TEXT -1 4 ". \+ " }}{PARA 15 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 271 11 "arctan(y,x) " }{TEXT -1 83 " function with 2 arguments is designed to produce exac tly what is needed for theta." }}{PARA 15 "" 0 "" {TEXT -1 89 "These f unctions return floating point decimal numbers if the input contains a ny decimals." }}{PARA 15 "" 0 "" {TEXT -1 32 "These functions are part of the " }{TEXT 265 7 "VecCalc" }{TEXT -1 71 " package, and so can be used by name only after performing the command " }{TEXT 266 13 "with( VecCalc)" }{TEXT -1 4 " or " }{TEXT 267 22 "with(VecCalc,function)" } {TEXT -1 58 ". The functions can always be accessed in the long forms " }{TEXT 268 17 "VecCalc[function]" }{TEXT -1 61 ". The aliases can \+ be used only after performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "E xamples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "polar2rec t([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"aG\"\"\"-%$cosG6#% \"bGF&*&F%F&-%$sinGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " rect2polar();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")S O\"H\"-%'VECTORG6#7$*$,&*$)%\"aG\"\"#\"\"\"F1*$)%\"bGF0F1F1#F1F0-%'arc tanG6$F4F/&%'VectorG6#%$rowG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" } {TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D " "" }{TEXT 26 2 ", " }{HYPERLNK 17 "AngleConversion" 2 "AngleConversi on" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12913640 }{RTABLE M7R0 I5RTABLE_SAVE/12913640X*%)anythingG6"6"[gl!$%!!!"#"#*$,&*$%"aG""#"""*$%"bGF+F,# F,F+-%'arctanG6$F.F*6" } c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "polar2rec t([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"aG\"\"\"-%$cosG6#% \"bGF&*&F%F&-%$sinGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " rect2polar();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")S O\"H\"-%'VECTORG6#7$*$,&*$)%\"aG\"\"#\"\"\"F1*$)%\"bGF0F1F1#F1F0-%'arc tanG6$F4F/&%'VectorG6#%$rowG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" } {TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D " "" }{TEXT 26 2 ", " }{HYPERLNK 17 "AngleConversion""ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadcouriercouriernormalbulletitemfunctveccalcjacobianmatrixcalculatjacobianmatrixcoordinattransformatparametrizedsurfacspacaliacanusedafterexecutvcaliacommandjaccallsequencjacobianmatrixvarsouttypjacobianmatrixparameterformlistvectorarrowdefinfunctionvariabloptionalnameindependariabltypelistlistdescriptrgkgngwhosijthntrypartialderivatcomponwithrespectactsfunctionreturnfirstderivativspecificatoptionalunlesunabldeterminhappenconstantbuiltundefinparametspecificonverthaveoutotherwisproducproducdifferlinalgjacobianpackagexpressrequiresvectorcalculuvectorcalculusexpressionpackagebutvectoralsocomputdeterminantpartndonlyperformveccalalwayaccesslongexamplmakefuncttgrtablegopugvgoperatorgarrowgfvectorgcolumngjtevalfunctjtghmmatrixgpvagfbgfmfrhothetaphisincosnrhosgrhogthetagphigsingcosgjspi"Yibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfixewidthfunctionveccalcsurfaceintscalarinertdisplaysurfacintegralscalarfieldsurfaceintscalarcomputesbothcommandcandisplaintermediatstepaliasusedafterxecutvcaliavcsiscallsequencvarrngsurfaceintscalarinertsurfaceintscalarinertarsurfaceintscalarparameterfunctvariablarrownotationparametricformlistdefinintegratcurvrangoveroptionalparametindicatthatdescriptacefirstargumsecondthirdfourthgumenttheirintegralevaluatusingvaluevalfcalculatwithoutincludwhilallintermediatreturninertmultipleintscalaryoudoneedquotesaroundassignariablappeareithorderwhatevbesthowevnamesmustmatchnameparamtersdefinittheydifferntialsimilarvectorcalculuaceintsurfaceintpackagewithdomainchosenusesexpressinsteadcannotnorshowstepsthespartpackagonlyperformalwayaccesslongsedvcaliasexamplesakefunctfgfxgygzg CurveTangent,VecCalc"#CurveTangentialAcceleration,VecCalc"CurveTorsion,VecCalc"CurveVelocity,VecCalc" Cv,VecCalc" D,VecCalc" Det,VecCalc"Determinant,VecCalc" Diff,VecCalc" Div,VecCalc"Divergence,VecCalc" Dot,VecCalc" EF,VecCalc" Grad,VecCalc"Gradient,VecCalc" Hess,VecCalc"Hessian,VecCalc" Int,VecCalc" JDet,VecCalc" Jac,VecCalc" rectangular "" recurrsive"red"referenc"region"rela"releas"remem"rememb """"renthes"repeat"repres" represent""""req"requir3""""""""""""reserv"reset"resolv"ress"restrict """result""""" MathematicsMathematics/PackagesMathematics/Packages/VecCalc%Mathematics/Packages/VecCalc/Aliases2#Mathematics/Packages/VecCalc/Angle Angle-Mathematics/Packages/VecCalc/AngleConversionAngleConversion&Mathematics/Packages/VecCalc/CommandsI/Mathematics/Packages/VecCalc/CoordConversion2DCoordConversion2D/Mathematics/Packages/VecCalc/CoordConversion3DCoordConversion3D#Mathematics/Packages/VecCalc/Cross Cross"Mathematics/Packages/VecCalc/Curl Curl#Mathematics/Packages/VecCalc/Curve Curve)Mathematics/Packages/VecCalc/CurveForgetCurveForget)Mathematics/Packages/VecCalc/DeterminantDeterminant"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYL"4<{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed \+ Width" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 48 " \+ 3 Dimensional Coordinate Conversions using the " }{TEXT 274 7 "VecCalc " }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT 26 11 "Functions: " } {TEXT -1 1 "\n" }{TEXT 260 23 " VecCalc[cyl2rect] - " }{TEXT -1 52 " Converts Coordinates from Cylindrical to Rectangular" }}{PARA 0 "" 0 " " {TEXT 261 23 " VecCalc[rect2cyl] - " }{TEXT -1 52 "Converts Coordi nates from Rectangular to Cylindrical" }}{PARA 0 "" 0 "" {TEXT 258 23 " VecCalc[sph2rect] - " }{TEXT -1 50 "Converts Coordinates from Sphe rical to Rectangular" }}{PARA 0 "" 0 "" {TEXT 259 23 " VecCalc[rect2 sph] - " }{TEXT -1 50 "Converts Coordinates from Rectangular to Spheri cal" }}{PARA 0 "" 0 "" {TEXT 256 23 " VecCalc[sph2cyl] - " }{TEXT -1 50 "Converts Coordinates from Spherical to Cylindrical" }}{PARA 0 " " 0 "" {TEXT 257 23 " VecCalc[cyl2sph] - " }{TEXT -1 50 "Converts C oordinates from Cylindrical to Spherical" }}{PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliases can be used after execution \+ of the " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command. " }}{PARA 257 "" 0 "" {TEXT -1 19 " c2r = cyl2rect" }}{PARA 257 " " 0 "" {TEXT -1 19 " r2c = rect2cyl" }}{PARA 257 "" 0 "" {TEXT -1 19 " s2r = sph2rect" }}{PARA 257 "" 0 "" {TEXT -1 19 " r2s = r ect2sph" }}{PARA 257 "" 0 "" {TEXT -1 18 " s2c = sph2cyl" }}{PARA 257 "" 0 "" {TEXT -1 18 " c2s = cyl2sph" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 80 " cyl2rect([r,theta,z]) c2r([r,theta,z]) VecCalc[cyl2r ect]([r,theta,z])" }}{PARA 256 "" 0 "" {TEXT -1 76 " rect2cyl([x,y,z ]) r2c([x,y,z]) VecCalc[rect2cyl]([x,y,z])" }}{PARA 256 "" 0 "" {TEXT -1 84 " sph2rect([rho,theta,phi]) s2r([rho,theta,p hi]) VecCalc[sph2rect]([rho,theta,phi])" }}{PARA 256 "" 0 "" {TEXT -1 76 " rect2sph([x,y,z]) r2s([x,y,z]) VecCalc[rect2sph ]([x,y,z])" }}{PARA 256 "" 0 "" {TEXT -1 83 " sph2cyl([rho,theta,phi ]) s2c([rho,theta,phi]) VecCalc[sph2cyl]([rho,theta,phi])" }}{PARA 256 "" 0 "" {TEXT -1 79 " cyl2sph([r,theta,z]) c2s([r,theta,z]) VecCalc[cyl2sph]([r,theta,z])" }}{PARA 0 "" 0 "" {TEXT 26 12 "Par ameters: " }}{PARA 0 "" 0 "" {TEXT 262 35 " [x,y,z] or or - " }{TEXT -1 24 "rectangular coordinates " }}{PARA 0 "" 0 "" {TEXT 263 18 " x - " }{TEXT -1 28 "first horizontal coord inate " }}{PARA 0 "" 0 "" {TEXT 264 18 " y - " }{TEXT -1 29 "second horizontal coordinate " }}{PARA 0 "" 0 "" {TEXT 265 18 " \+ z - " }{TEXT -1 66 "vertical axis, positive upward and rela ted by the right hand rule " }}{PARA 0 "" 0 "" {TEXT 266 47 " [r,the ta,z] or or - " }{TEXT -1 24 "cylindrical coor dinates " }}{PARA 0 "" 0 "" {TEXT 267 18 " r - " }{TEXT -1 43 "the perpendicular distance from the z-axis " }}{PARA 0 "" 0 "" {TEXT 268 18 " theta - " }{TEXT -1 72 "the angle measured in \+ radians counterclockwise from the positive x-axis " }}{PARA 0 "" 0 "" {TEXT 269 18 " z - " }{TEXT -1 22 "same as rectangular z \+ " }}{PARA 0 "" 0 "" {TEXT 270 59 " [rho,theta,phi] or or - " }{TEXT -1 22 "spherical coordinates " }} {PARA 7 "" 0 "" {TEXT -1 71 "CAUTION: The spherical coordinate system \+ used by Maple is left handed. " }}{PARA 0 "" 0 "" {TEXT 271 18 " rho - " }{TEXT -1 36 "the radial distance from the origin " }} {PARA 0 "" 0 "" {TEXT 272 18 " theta - " }{TEXT -1 26 "same a s cylindrical theta " }}{PARA 0 "" 0 "" {TEXT 273 18 " phi \+ - " }{TEXT -1 60 "the polar angle measured in radians from the positiv e z-axis" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 276 8 "cyl2rect" }{TEXT -1 98 " converts cylindrical coordinates to rectangular coordinates using th e formulas:\n " }{XPPEDIT 19 1 "x = r cos(theta);" "6# /%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 12 " " } {XPPEDIT 19 1 "y = r sin(theta) ;" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6#%&th etaGF'" }{TEXT -1 10 " " }{XPPEDIT 19 1 "z = z;" "6#/%\"zGF$ " }{TEXT -1 73 " \nThere are no restriction on the values of the cyli ndrical coordinates." }}{PARA 15 "" 0 "" {TEXT 277 8 "rect2cyl" } {TEXT -1 134 " converts rectangular coordinates to cylindrical coordin ates using the formulas:\n \+ " }{XPPEDIT 19 1 "theta = arctan(y/x);" "6#/%&thetaG-%'arctan G6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 42 " in quadrants I and IV \n \+ " }{XPPEDIT 19 1 "r = sqrt(x^2 + y^2) ;" "6#/%\"rG-%%sq rtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT -1 13 " " } {XPPEDIT 19 1 "theta = arctan(y/x) +Pi;" "6#/%&thetaG,&-%'arctanG6#*&% \"yG\"\"\"%\"xG!\"\"F+%#PiGF+" }{TEXT -1 37 " in quadrant II \+ " }{XPPEDIT 19 1 "z = z;" "6#/%\"zGF$" }{TEXT -1 57 " \n " }{XPPEDIT 19 1 "theta=arctan(y/x )-Pi;" "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG! \"\"F+%#PiGF-" }{TEXT -1 86 " in quadrant III \nThe resulting cylindr ical coordinates are restricted to the ranges " }{XPPEDIT 19 1 "r >= 0 ;" "6#1\"\"!%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 19 1 " -Pi < theta;" "6#2,$%#PiG!\"\"%&thetaG" }{XPPEDIT 19 1 "``<= Pi;" "6#1%!G%#PiG" } {TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 275 11 "arctan(y,x)" }{TEXT -1 83 " function with 2 arguments is designed \+ to produce exactly what is needed for theta." }}{PARA 15 "" 0 "" {TEXT 278 8 "sph2rect" }{TEXT -1 96 " converts spherical coordinates t o rectangular coordinates using the formulas:\n " } {XPPEDIT 19 1 "x = rho * sin(phi)*cos(theta);" "6#/%\"xG*(%$rhoG\"\"\" -%$sinG6#%$phiGF'-%$cosG6#%&thetaGF'" }{TEXT -1 12 " " } {XPPEDIT 19 1 "y = rho *sin(phi) *sin(theta) ;" "6#/%\"yG*(%$rhoG\"\" \"-%$sinG6#%$phiGF'-F)6#%&thetaGF'" }{TEXT -1 10 " " } {XPPEDIT 19 1 "z = rho *cos(phi);" "6#/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$p hiGF'" }{TEXT -1 71 " \nThere are no restriction on the values of the spherical coordinates." }}{PARA 15 "" 0 "" {TEXT 279 8 "rect2sph" } {TEXT -1 119 " converts rectangular coordinates to spherical coordinat es using the formulas:\n " } {XPPEDIT 19 1 "theta = arctan(y/x);" "6#/%&thetaG-%'arctanG6#*&%\"yG\" \"\"%\"xG!\"\"" }{TEXT -1 28 " in quadrants I and IV \n " } {XPPEDIT 19 1 "rho = sqrt(x^2 + y^2 + z^2) ;" "6#/%$rhoG-%%sqrtG6#,(*$ %\"xG\"\"#\"\"\"*$%\"yGF+F,*$%\"zGF+F," }{TEXT -1 6 " " } {XPPEDIT 19 1 "theta = arctan(y/x) +Pi;" "6#/%&thetaG,&-%'arctanG6#*&% \"yG\"\"\"%\"xG!\"\"F+%#PiGF+" }{TEXT -1 25 " in quadrant II \+ " }{XPPEDIT 19 1 "phi = arctan(sqrt(x^2+y^2)/z);" "6#/%$phiG-%'arctanG 6#*&-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF/F0F0%\"zG!\"\"" }{TEXT -1 41 "\n " }{XPPEDIT 19 1 "theta= arctan(y/x )-Pi;" "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+% #PiGF-" }{TEXT -1 84 " in quadrant III \nThe resulting spherical coor dinates are restricted to the ranges " }{XPPEDIT 19 1 "rho >= 0;" "6#1 \"\"!%$rhoG" }{TEXT -1 2 ", " }{XPPEDIT 19 1 " -Pi < theta; " "6#2,$%# PiG!\"\"%&thetaG" }{XPPEDIT 19 1 "``<= Pi;" "6#1%!G%#PiG" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "0 <= phi; " "6#1\"\"!%$phiG" }{XPPEDIT 19 1 " ``<=Pi;" "6#1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT 280 7 "sph2cyl" }{TEXT -1 96 " converts spherical coordinates to cylin drical coordinates using the formulas:\n " }{XPPEDIT 19 1 "r = rho * sin(phi);" "6#/%\"rG*&%$rhoG\"\"\"-%$sinG6#%$phiGF'" } {TEXT -1 12 " " }{XPPEDIT 19 1 "theta = theta;" "6#/%&theta GF$" }{TEXT -1 13 " " }{XPPEDIT 19 1 "z = rho *cos(phi);" "6#/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF'" }{TEXT -1 86 " \nThere are \+ no restriction on the values of the spherical or cylindrical coordinat es." }}{PARA 15 "" 0 "" {TEXT 281 7 "cyl2sph" }{TEXT -1 96 " converts \+ cylindrical coordinates to spherical coordinates using the formulas:\n " }{XPPEDIT 19 1 "rho = sqrt(r^2 + z^2) ;" "6#/%$rhoG -%%sqrtG6#,&*$%\"rG\"\"#\"\"\"*$%\"zGF+F," }{TEXT -1 9 " " } {XPPEDIT 19 1 "theta = theta;" "6#/%&thetaGF$" }{TEXT -1 12 " \+ " }{XPPEDIT 19 1 "phi = arccos(z/sqrt(r^2 + z^2));" "6#/%$phiG-%'ar ccosG6#*&%\"zG\"\"\"-%%sqrtG6#,&*$%\"rG\"\"#F**$F)F1F*!\"\"" }{TEXT -1 140 " \nThere are no restriction on the values of the cylindrical coordinates.\nThe resulting spherical coordinates are restricted to t he ranges " }{XPPEDIT 19 1 "rho >= 0;" "6#1\"\"!%$rhoG" }{TEXT -1 6 " \+ and " }{XPPEDIT 19 1 "0 <= phi; " "6#1\"\"!%$phiG" }{XPPEDIT 19 1 "`` <=Pi;" "6#1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 89 "These functions return floating point decimal numbers if the input contains any decimals." }}{PARA 15 "" 0 "" {TEXT -1 32 "These functio ns are part of the " }{TEXT 282 7 "VecCalc" }{TEXT -1 71 " package, an d so can be used by name only after performing the command " }{TEXT 283 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 284 22 "with(VecCalc, function)" }{TEXT -1 58 ". The functions can always be accessed in th e long forms " }{TEXT 285 17 "VecCalc[function]" }{TEXT -1 61 ". The \+ aliases can be used only after performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cyl2rect([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"a G\"\"\"-%$cosG6#%\"bGF&*&F%F&-%$sinGF)F&%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2cyl();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?P\"H\"-%'VECTORG6#7%*$,&*$)%\"aG\"\"#\"\"\"F1*$) %\"bGF0F1F1#F1F0-%'arctanG6$F4F/%\"cG&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sph2rect();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")gP\"H\"-%'VECTORG6#7%*(%\"aG\"\"\"- %$sinG6#%\"cGF--%$cosG6#%\"bGF-*(F,F-F.F--F/F4F-*&F,F--F3F0F-&%'Vector G6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2sph([a,b, c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,(*$)%\"aG\"\"#\"\"\"F**$) %\"bGF)F*F**$)%\"cGF)F*F*#F*F)-%'arctanG6$F-F(-F36$*$,&F&F*F+F*F1F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sph2cyl();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+Q\"H\"-%'MATRIXG6#7%7# *&%\"aG\"\"\"-%$sinG6#%\"cGF.7#%\"bG7#*&F-F.-%$cosGF1F.&%'VectorG6#%'c olumnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "cyl2sph(); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!)Q\"H\"-%'VECTORG6 #7%*$,&*$)%\"aG\"\"#\"\"\"F1*$)%\"cGF0F1F1#F1F0%\"bG-%'arctanG6$F/F4&% 'VectorG6#%$rowG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Copyright 1 995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "Se e Also: " }{HYPERLNK 17 "VecCalc" 2 "vec_calc" "" }{TEXT 26 2 ", " } {HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "AngleConversion" 2 "AngleConversion" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12913720 12913760 12913800 12913880 }{RTABLE M7R0 I5RTABLE_SAVE/12913720X*%)anythingG6"6"[gl!$%!!!"$"$*$,&*$%"aG""#"""*$%"bGF+F,# F,F+-%'arctanG6$F.F*%"cG6" } {RTABLE M7R0 I5RTABLE_SAVE/12913760X*%)anythingG6"6"[gl!$%!!!"$"$*(%"aG"""-%$sinG6#%"cGF)-%$ cosG6#%"bGF)*(F(F)F*F)-F+F0F)*&F(F)-F/F,F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12913800X*%)anythingG6"6"[gl!#%!!!"$"$*&%"aG"""-%$sinG6#%"cGF)%"b G*&F(F)-%$cosGF,F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12913880X*%)anythingG6"6"[gl!$%!!!"$"$*$,&*$%"aG""#"""*$%"cGF+F,# F,F+%"bG-%'arctanG6$F*F.6" } 115 "Copyright 1 995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "Se e Also: " }{HYPERLNK 17 "VecCalc" 2 "vec_calc" "" }{TEXT 26 2 ", " } {HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT 26 2 VecCalc,rect2sph" VecCalc,s2c" VecCalc,s2r"VecCalc,simplifyvec" VecCalc,sis" VecCalc,siv"VecCalc,sph2cyl"VecCalc,sph2rect"VecPot"VectorPotential"`&->`"`&.`"`&@`"`&x`"angle"c2r"c2s"cross"curl"curve"cyl2rect"cyl2sph"d2r"deg2rad"det" determinant"diff"diffops"div" divergence"dot" evalFunction"spotg"sq"sqrt"""sqrtg ""squar""""ss ""st """"""ste ""stepG" """"stg"still"stlist"stor ""structur"stud""" su"""" submatric" submatrix"subs"subsequ""" substitut"such""""sum"phig """"phigf ""philip#""""""""""""""""""""""""""""""""""pi/*""" """"""""pics"pig'"""""""""pigf'"""""""""ple"plot""""plott ""po"""point#(""""""""pointg"polar"""" polynomial"posit """"positiv"possib"possibl"potent" potential"" "pr" precedenc""""prev"previou""ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfunctveccalclaplaciancalculataplacianlistfunctionarrownotataliacanusedafterexecutvcaliacommandlaplaplaciancallsequencfncvarsarsparameterdefinpossibnestlisvectormatrixarraoptionanameindependvariabldescriptargumsinglreturnformrecurrsivemapsontentrspecificatoptionalunlesunabldeterminhappenconstantbuiltundefindifferlinalgpackagactsexpresscannotactlargarrayrequirvariablebutworknoncartesiancoordinatvectorcalculuvectorcalculuvectorfieldvectorcalculusystemfunctionpartthonlyperformwithalwayaccesslonglaplacianlapbperforminexamplfgfxgygzgoperatorgarrowgflflfgfoperatorgbgfagfcgfmakefunctfgrtablegovlmatrixgfcfvectorgcolumnglapfeflapflapfgepfaffartablegwlcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoeccalclinallaplaciandiffopgradidivergenchessianrtabl".ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriercouriernormalbulletitemfixedwidthourifunctveccalcvectorpotentialcalculatvectorpotentialfieldarrownotatexistaliacausedafterexecutvcaliacommandvpotcallsequencvarsouttypparameterformlistdefinfunctionvariabloptionalnameindependtyperowcolumndescrippotentialfieldwhoscurlactsvectorfunctionreturnvectordoesundefinhespecificatunlesunabldetermincanhappenfunctionconstantbuiltparametspecificonverthaveotherwismatchvectorpotentialdifferlinalgvecpotpackagexpressrequirtruefalsextraargumencommandvectorcalculuvectorcalculusvectorcalculuvectorfieldnullcoordinatsystempartonlyperformwithtorpotentialalwayaccesslongexamplveccalmakefuncagxgygzgoperatorgarrowgfcgoperatorgrtablegmatrixgfdfvectorgcolumngundefinedgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoveccalcecpotvectorpotentialdCurveSpeed,VecCalc Jac,VecCalcOutputVectorType,VecCalcycyl2sph,VecCalcsingfcgrtablegvectorgagrowggpcgfvectormatrixgbgcosgfolumngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitsealsoveccalccoordconversangleconversrtablsaveanythinggglectangularcoordinatesphsphericalsphericalsphericaloordinataliascanusedafterexecutvcaliacommandectcallsequencthetarhophihiparameterfirsthorizontalcoordinatsecondverticalaxispositupwardrelatedrighthandruletacoordinatperpendiculardistancanglmeasurradiancounterclockwissamecautsystemleftradialoriginpolarpositivdescriptthformulacosxgrgcosgthetagfsinygsingetagfzgfnthernorestrictvalucylindricalcoordinatesarctanthetagquadrantivsqrtsqrtgygfpiarctangpigfiiiiintheresultcylindricalrangpigfunctwithargumentdesignproducexactneedrhogphigfzghigfessqrtgphigcylindricalgfarccoarccosghethesreturnfloatpointdecimalnumbercontainanyfunctionspartnameonlyperformalwayaccesslongformexamplbgf"veccalc"VecCalc,VCaliasvcalias"TVecCalc,OutputMatrixTypeVecCalc,OutputVectorTypeOutputMatrixTypeOutputVectorType"VecCalc,simplifyvec"xVecCalc,DVecCalc,intVecCalc,IntVecCalc,diffVecCalc,DiffVecCalc,limitVecCalc,LimitDintIntdiffDifflimitLimit"9VecCalc,&->VecCalc,MFVecCalc,MakeFunctionMF`&->`&->"6VecCalc,&@VecCalc,EFVecCalc,evalFunctionEF`&@`&@"#VecCalc,&.VecCalc,Dotdot`&.`&."VecCalc,Lengthlength)Mathematics/Packages/VecCalc/Commands/&@evalFunction)Mathematics/Packages/VecCalc/Commands/&x Cross,Mathematics/Packages/VecCalc/Commands/Angle Angle6Mathematics/Packages/VecCalc/Commands/AngleConversionAngleConversion8Mathematics/Packages/VecCalc/Commands/CoordConversion2DCoordConversion2D8Mathematics/Packages/VecCalc/Commands/CoordConversion3DCoordConversion3D,Mathematics/Packages/VecCalc/Commands/Cross Cross+Mathematics/Packages/VecCalc/Commands/Curl Curl8Mathematics/Packages/VecCalc/Commands/CurveAcceleration Curve""""",Mathematics/Packages/VecCalc/Commands/limitMappedFunctions2Mathematics/Packages/VecCalc/Commands/multipleintMultipleint1Mathematics/Packages/VecCalc/Commands/polar2rectCoordConversion2D.Mathematics/Packages/VecCalc/Commands/rad2degAngleConversion/Mathematics/Packages/VecCalc/Commands/rect2cylCoordConversion3D1Mathematics/Packages/VecCalc/Commands/rect2polarCoordConversion2D/Mathematics/Packages/VecCalc/Commands/rect2sphCoordConversion3D2Mathematics/Packages/VecCalc/Commands/simplifyvecsimplifyvec.Mathematics/Packages/VecCalc/Commands/sph2cylCoordConversion3D/Mathematics/Packages/VecCalc/Commands/sph2rectCoordConversion3DCoordConversion3D-Mathematics/Packages/VecCalc/Commands/sph2cylCross"Mathematics/Packages/VecCalc/CrossCross+Mathematics/Packages/VecCalc/Commands/CrossCross(Mathematics/Packages/VecCalc/Commands/&xCurl!Mathematics/Packages/VecCalc/CurlCurl*Mathematics/Packages/VecCalc/Commands/CurlCurve"Mathematics/Packages/VecCalc/CurveCurve3Mathematics/Packages/VecCalc/Commands/CurveVelocityCurve7Mathematics/Packages/VecCalc/Commands/CurveAccelerationCurve/Mathematics/Packages/VecCalc/Commands/CurveJerkCurve2Mathematics/Packages/VecCalc/Commands/CurveTangentCurve'Mathematics/Packages/VecCalc/Aliases/CBCurve'Mathematics/Packages/VecCalc/Aliases/CkCurve'Mathematics/Packages/VecCalc/Aliases/CtCurve'Mathematics/Packages/VecCalc/Aliases/CsCurve'Mathematics/Packages/VecCalc/Aliases/CLCurve(Mathematics/Packages/VecCalc/Aliases/CaTCurve(Mathematics/Packages/VecCalc/Aliases/CaN} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 53 " \+ Frenet Analysis of a Curve using the VecCalc Package" }}{PARA 0 "" 0 " " {TEXT 26 18 "Calling Sequences:" }}{PARA 256 "" 0 "" {TEXT -1 52 " \+ command(r,t) alias(r,t) VecCalc[command](r,t)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT 256 15 " command \+ - " }{TEXT -1 29 "a command from the list below" }}{PARA 0 "" 0 "" {TEXT 278 15 " alias - " }{TEXT -1 28 "an alias from the list be low" }}{PARA 0 "" 0 "" {TEXT 257 15 " r - " }{TEXT -1 73 "a \+ curve in the form of a list or Vector of expressions with one paramete r" }}{PARA 0 "" 0 "" {TEXT 272 15 " " }{TEXT -1 42 "or a n expression which simplifies to such." }}{PARA 0 "" 0 "" {TEXT 271 15 " t - " }{TEXT -1 27 "the parameter for the curve" }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 156 "These commands are designed to perform a Frenet analysis of a curve. Each command has a shorter alias. Most commands work in any dimension. However, the " }{TEXT 273 13 "CurveBinormal" }{TEXT -1 5 " and " }{TEXT 274 12 "CurveTorsion" }{TEXT -1 40 " commands only work in three dimensions." }}{PARA 15 "" 0 "" {TEXT -1 95 "Below is a list of each command, the corresponding alias and a short description of its action." }}{PARA 0 "" 0 "" {TEXT 258 41 " CurveVelocity \+ Cv - " }{TEXT -1 23 "Calculate the Velocity " }}{PARA 0 "" 0 "" {TEXT 259 41 " CurveAcceleration Ca - " } {TEXT -1 27 "Calculate the Acceleration " }}{PARA 0 "" 0 "" {TEXT 260 41 " CurveJerk Cj - " }{TEXT -1 19 "Calculate the Jerk " }}{PARA 0 "" 0 "" {TEXT 269 41 " CurveSpeed \+ Cs - " }{TEXT -1 20 "Calculate the Speed " }}{PARA 0 "" 0 " " {TEXT 270 41 " CurveArcLength CL - " }{TEXT -1 25 "Calculate the Arc Length " }}{PARA 0 "" 0 "" {TEXT 261 41 " Curve Tangent CT - " }{TEXT -1 27 "Calculate the Unit T angent " }}{PARA 0 "" 0 "" {TEXT 262 41 " CurveNormal \+ CN - " }{TEXT -1 36 "Calculate the Unit Principal Normal " }} {PARA 0 "" 0 "" {TEXT 263 41 " CurveBinormal CB - " }{TEXT -1 38 "Calculate the Unit Binormal (3D only) " }}{PARA 0 "" 0 "" {TEXT 264 41 " CurveCurvature Ck - " }{TEXT -1 24 "Calculate the Curvature " }}{PARA 0 "" 0 "" {TEXT 265 41 " Cur veTorsion Ct - " }{TEXT -1 32 "Calculate the Tors ion (3D only) " }}{PARA 0 "" 0 "" {TEXT 266 41 " CurveTangentialAccel eration CaT - " }{TEXT -1 38 "Calculate the Tangential Accelerati on " }}{PARA 0 "" 0 "" {TEXT 267 41 " CurveNormalAcceleration Ca N - " }{TEXT -1 34 "Calculate the Normal Acceleration " }}{PARA 0 "" 0 "" {TEXT 268 41 " CurveForget Cforget - " } {TEXT -1 49 "Clear the remember tables for the above commands " }} {PARA 15 "" 0 "" {TEXT -1 48 "To use the command name, you must first \+ execute " }{TEXT 275 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 276 21 "with(VecCalc,command)" }{TEXT -1 58 ". To use the aliases, you mu st first execute the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 206 "The curve must be in \+ the form of a list or Vector of expressions with one parameter, except that Maple will first attempt to simplify it to this form. Such a cu rve may be plotted in 2 dimensions using the " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 65 " command with a parametric argument or in 3 di mensions using the " }{HYPERLNK 17 "spacecurve" 2 "spacecurve" "" } {TEXT -1 18 " command from the " }{HYPERLNK 17 "plots" 2 "plots" "" } {TEXT -1 9 " package." }}{PARA 15 "" 0 "" {TEXT -1 20 "Each command us es a " }{HYPERLNK 17 "remember" 2 "remember" "" }{TEXT -1 170 " table \+ to speed up the computation. These tables may be cleared after finishi ng the Frenet analysis of a curve to avoid cluttering the memory. Thi s is done by using the " }{HYPERLNK 17 "CurveForget" 2 "CurveForget" " " }{TEXT -1 18 " command from the " }{TEXT 279 7 "VecCalc" }{TEXT -1 22 " package or its alias " }{TEXT 277 7 "Cforget" }{TEXT -1 1 "." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Examples: " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A 2D Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R:=;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"RG-%'RTABLEG6%\")?V\"H\"-%'VECTORG6#7$*&%\"tG\"\"\"-%$cosG6#F.F/*&F. F/-%$sinGF2F/&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([R[1],R[2],t=-2*Pi..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "CurveVelocity(R,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\")gV\"H\"-%'VECTORG6#7$,&-%$cosG6#%\"tG\"\"\"*&F/F0-%$s inGF.F0!\"\",&F2F0*&F/F0F,F0F0&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "CurveAcceleration(R,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")SW\"H\"-%'VECTORG6#7$,&*&\"\"#\"\"\"-%$s inG6#%\"tGF.!\"\"*&F2F.-%$cosGF1F.F3,&*&F-F.F5F.F.*&F2F.F/F.F3&%'Vecto rG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "CurveJerk(R,t ); #(The derivative of the acceleration.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")![9H\"-%'VECTORG6#7$,&*&\"\"$\"\"\"-%$co sG6#%\"tGF.!\"\"*&F2F.-%$sinGF1F.F.,&*&F-F.F5F.F3*&F2F.F/F.F3&%'Vector G6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "CurveSpeed(R,t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)%\"tG\"\"#\"\"\"F)F)F)#F) F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "L:=CurveArcLength(R,t ); L(0,2*Pi); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6 $%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG6$*$,&*$)%\"tG\"\"#\"\"\"F 6F6F6#F6F5/F4;9$9%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$* $,&*$)%\"tG\"\"#\"\"\"F,F,F,#F,F+/F*;\"\"!,$*&F+F,%#PiGF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#PiG\"\"\",&*&\"\"%F&)F%\"\"#F&F&F&F&#F &F+F&*&#F&F+F&-%#lnG6#,&*&F+F&F%F&!\"\"*$F'F,F&F&F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "CurveTangent(R,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?X\"H\"-%'VECTORG6#7$,$*&,&-%$cosG6#%\"t G!\"\"*&F1\"\"\"-%$sinGF0F4F4F4,&*$)F1\"\"#F4F4F4F4#F2F:F2*&F7F;,&F5F4 *&F1F4F.F4F4F4&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "CurveNormal(R,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'RTABLEG6%\")gX\"H\"-%'VECTORG6#7$,$*&,&*$)%\"tG\"\"#\"\"\"F2F2F2#! \"\"F1,&-%$sinG6#F0F2*&F0F2-%$cosGF8F2F2F2F4,$*&,&F:F4*&F0F2F6F2F2F2F- F3F4&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C urveCurvature(R,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"tG\" \"#\"\"\"F)F(F)F),&F%F)F)F)#!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "CurveTangentialAcceleration(R,t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&%\"tG\"\"\",&*$)F$\"\"#F%F%F%F%#!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "CurveNormalAcceleration(R,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"tG\"\"#\"\"\"F)F(F)F),&F%F)F )F)#!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "CurveForget( R);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A 3D Example using aliases :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r:=;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'RTABLEG6%\")+Y\"H\"-%' VECTORG6#7%*&%\"tG\"\"\"-%$cosG6#F.F/*&F.F/-%$sinGF2F/F.&%'VectorG6#%$ rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "spacecurve([r[1],r[ 2],r[3],t=-2*Pi..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " Cv(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!o9H\"-%'VE CTORG6#7%,&-%$cosG6#%\"tG\"\"\"*&F/F0-%$sinGF.F0!\"\",&F2F0*&F/F0F,F0F 0F0&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Ca( r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")7o;6-%'VECTORG 6#7%,&*&\"\"#\"\"\"-%$sinG6#%\"tGF.!\"\"*&F2F.-%$cosGF1F.F3,&*&F-F.F5F .F.*&F2F.F/F.F3\"\"!&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Cj(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG 6%\")gZ\"H\"-%'VECTORG6#7%,&*&\"\"$\"\"\"-%$cosG6#%\"tGF.!\"\"*&F2F.-% $sinGF1F.F.,&*&F-F.F5F.F3*&F2F.F/F.F3\"\"!&%'VectorG6#%$rowG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Cs(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$)%\"tG\"\"#\"\"\"F)F(F)#F)F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "L:=CL(r,t); L(0,2*Pi); value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6$%\"aG%\"bG6\"6$%)operatorG%& arrowGF)-%$IntG6$*$,&*$)%\"tG\"\"#\"\"\"F6F5F6#F6F5/F4;9$9%F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&*$)%\"tG\"\"#\"\"\"F,F+F, #F,F+/F*;\"\"!,$*&F+F,%#PiGF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &%#PiG\"\"\",&*&\"\"%F&)F%\"\"#F&F&F+F&#F&F+F&-%#lnG6#,&*&F+F,F%F&F&*$ ,&*&F+F&F*F&F&F&F&F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " CT(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+[\"H\"-%'V ECTORG6#7%,$*&,&-%$cosG6#%\"tG!\"\"*&F1\"\"\"-%$sinGF0F4F4F4,&*$)F1\" \"#F4F4F:F4#F2F:F2*&F7F;,&F5F4*&F1F4F.F4F4F4*&F4F4*$F7#F4F:F2&%'Vector G6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CN(r,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")S[\"H\"-%'VECTORG6#7%,$ *(,&*$)%\"tG\"\"#\"\"\"F2F1F2#!\"\"F1,(*$)F0\"\"%F2F2*&\"\"&F2F/F2F2\" \")F2F3,**&-%$sinG6#F0F2F/F2F2*&F8F2F>F2F2*&)F0\"\"$F2-%$cosGF@F2F2*(F DF2F0F2FEF2F2F2F4*(,**&FEF2F/F2F2*&F8F2FEF2F2*&FCF2F>F2F4*(FDF2F0F2F>F 2F4F2F-F3F5F3,$*(F-F3F5F3F0F2F4&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "CB(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'RTABLEG6%\")g\\\"H\"-%'VECTORG6#7%*&,(*$)%\"tG\"\"%\"\"\"F1*&\"\"&F1 )F/\"\"#F1F1\"\")F1#!\"\"F5,&*&F5F1-%$cosG6#F/F1F8*&F/F1-%$sinGF=F1F1F 1,$*&,&*&F5F1F?F1F1*&F/F1F;F1F1F1F,F7F8*&F,F7,&*$F4F1F1F5F1F1&%'Vector G6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Ck(r,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"tG\"\"%\"\"\"F)*&\"\"&F))F' \"\"#F)F)\"\")F)#F)F-,&*$F,F)F)F-F)#!\"$F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "Ct(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\" '\"\"\"*$)%\"tG\"\"#F&F&F&,(*$)F)\"\"%F&F&*&\"\"&F&F(F&F&\"\")F&!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "CaT(r,t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&%\"tG\"\"\",&*$)F$\"\"#F%F%F)F%#!\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "CaN(r,t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,(*$)%\"tG\"\"%\"\"\"F)*&\"\"&F))F'\"\"#F)F)\"\")F) #F)F-,&*$F,F)F)F-F)#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Cforget(r);" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright \+ 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department o f Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "Se e Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "CurveForget" 2 "CurveForget" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface" 2 "Surface" "" }{TEXT -1 1 "." }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12914320 12914360 12914440 12914480 12914520 12914560 12914600 12914680 11166812 12914760 12914800 12914840 12914960 }{RTABLE M7R0 I5RTABLE_SAVE/12914320X*%)anythingG6"6"[gl!$%!!!"#"#*&%"tG"""-%$cosG6#F(F)*&F(F )-%$sinGF,F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12914360X*%)anythingG6"6"[gl!$%!!!"#"#,&-%$cosG6#%"tG"""*&F+F,-%$ sinGF*F,!"",&F.F,*&F+F,F(F,F,6" } {RTABLE M7R0 I5RTABLE_SAVE/12914440X*%)anythingG6"6"[gl!$%!!!"#"#,&-%$sinG6#%"tG!"#*&F+"""-% $cosGF*F.!"",&F/""#*&F+F.F(F.F16" } {RTABLE M7R0 I5RTABLE_SAVE/12914480X*%)anythingG6"6"[gl!$%!!!"#"#,&-%$cosG6#%"tG!"$*&F+"""-% $sinGF*F.F.,&F/F,*&F+F.F(F.!""6" } {RTABLE M7R0 I5RTABLE_SAVE/12914520X*%)anythingG6"6"[gl!$%!!!"#"#,$*&,&-%$cosG6#%"tG!""*&F-" ""-%$sinGF,F0F0F0,&*$F-""#F0F0F0#F.F5F.*&F3F6,&F1F0*&F-F0F*F0F0F06" } {RTABLE M7R0 I5RTABLE_SAVE/12914560X*%)anythingG6"6"[gl!$%!!!"#"#,$*&,&*$%"tG""#"""F-F-#!""F ,,&-%$sinG6#F+F-*&F+F--%$cosGF3F-F-F-F/,$*&,&F5F/*&F+F-F1F-F-F-F)F.F/6" } {RTABLE M7R0 I5RTABLE_SAVE/12914600X*%)anythingG6"6"[gl!$%!!!"$"$*&%"tG"""-%$cosG6#F(F)*&F(F )-%$sinGF,F)F(6" } {RTABLE M7R0 I5RTABLE_SAVE/12914680X*%)anythingG6"6"[gl!$%!!!"$"$,&-%$cosG6#%"tG"""*&F+F,-%$ sinGF*F,!"",&F.F,*&F+F,F(F,F,F,6" } {RTABLE M7R0 I5RTABLE_SAVE/11166812X*%)anythingG6"6"[gl!$%!!!"$"$,&-%$sinG6#%"tG!"#*&F+"""-% $cosGF*F.!"",&F/""#*&F+F.F(F.F1""!6" } {RTABLE M7R0 I5RTABLE_SAVE/12914760X*%)anythingG6"6"[gl!$%!!!"$"$,&-%$cosG6#%"tG!"$*&F+"""-% $sinGF*F.F.,&F/F,*&F+F.F(F.!""""!6" } {RTABLE M7R0 I5RTABLE_SAVE/12914800X*%)anythingG6"6"[gl!$%!!!"$"$,$*&,&-%$cosG6#%"tG!""*&F-" ""-%$sinGF,F0F0F0,&*$F-""#F0F5F0#F.F5F.*&F3F6,&F1F0*&F-F0F*F0F0F0*$F3F66" } {RTABLE M7R0 I5RTABLE_SAVE/12914840X*%)anythingG6"6"[gl!$%!!!"$"$,$*(,&*$%"tG""#"""F,F-#!""F ,,(*$F+""%F-F*""&"")F-F.,**&-%$sinG6#F+F-F+F,F-F7F2*&F+""$-%$cosGF9F-F-*&F+F-F< F-F;F-F/*(,**&F" VecCalc,&." &->,VecCalc" &.,VecCalc" &@,VecCalc" &x,VecCalc" Angle,VecCalc" CB,VecCalc" CL,VecCalc" CN,VecCalc" CT,VecCalc" Ca,VecCalc" CaN,VecCalc" CaT,VecCalc"Cforget,VecCalc" Cj,VecCalc" Ck,VecCalc" Cross,VecCalc" Cs,VecCalc" Ct,VecCalc" Curl,VecCalc"CurveAcceleration,VecCalc"CurveArcLength,VecCalc"CurveBinormal,VecCalc"CurveCurvature,VecCalc"CurveForget,VecCalc"CurveJerk,VecCalc"CurveNormal,VecCalc"CurveNormalAcceleration,VecCalc"CurveSpeed,VecCalc"VecCalc,CurveForget"VecCalc,CurveJerk"VecCalc,CurveNormal"VecCalc,CurveNormalAcceleration"VecCalc,CurveSpeed"VecCalc,CurveTangent"#VecCalc,CurveTangentialAcceleration"VecCalc,CurveTorsion"VecCalc,CurveVelocity" VecCalc,Cv" VecCalc,D" VecCalc,Det"VecCalc,Determinant" VecCalc,Diff" VecCalc,Div"VecCalc,Divergence" VecCalc,Dot" VecCalc,EF" VecCalc,Grad"VecCalc,Gradient" VecCalc,Hess"VecCalc,Hessian" VecCalc,Int" VecCalc,JDet" VecCalc,Jac"VecCalc,JacobianDeterminant"VecCalc,JacobianMatrix" VecCalc,LPMD" VecCalc,Lap"VecCalc,Laplacian")VecCalc,LeadingPrincipalMinorDeterminants"VecCalc,Length""ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemtimesfixedwidthfunctionveccalclineintscalarinertisplaylineintegralscalarfieldlineintscalarcomputlinebothcommandcandisplaintermediatstepliasaliasusedafterexecuthevcaliavccommandlislineintscalarinertcallsequencneintscalarinertvarrnglineintscalarinertparameterfunctvariablarrownotatcurvformlistdefinparametintegratrangoveroptionalindicatdisplayeddescriptdisplayfirstargumsecondththirdintegralevaluatusingvaluevalfcalculatintegralwithoutincludscalarwhilallreturnsinertmultiplreturnyoudoneedquotaroundenassignsimilarvectorcalculuspathintvectorcalculupackagewithdomainchosenpathhowevusesexpressinsteadfunctioncannotnorshowthespartpackagnameonlyperformalwayaccesslonglyexamplveccalcmakefunfgfxgygzgoperatorgarrowgfmakefunctionsincosrgtgsingcosgflineintscalarinrtpiinttgfpigperatorgarrowgfintgpigfpiglngcurvetanggxsingurvecurvaturcurvetangentialacceleratvectorggzectorgdffeffcffdfcancopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitsealsosurfacrtablsaveanythinggglifsuchdescriptthesdesignperformeachshortmostworkanydimenshowevcurvebinormalcurvetorsonlycorrespondactioncurvevelocitcvcalculatvelocitcurveacceleratcaacceleratcurvejerkcjjerkcurvespecsspeedcurvearclengthclarclengthtangctunitangentcurvenormalcnprincipalcbbinormalcurvecurvaturckcurvaturcurvetorstorsioncurvetangentialacceleratcattangentialacceleraticurvenormalacceleratcurveforgetcforgetclearremembtablabovusenameyoumustfirstexecutaliasmustvcaliaexceptwillattemptcurvemayplottplotparametricargumdimensionspacecurvusesupcomputatafterfinishingavoidcluttermemorthidoneexamplcossinrgrtablegvectorgtgcosgsingfrowgpigvingfswingtgfcosgfvectoderivatcosgvalulgfagbgo"VecCalc,CaNVecCalc,CaTVecCalc,CLVecCalc,CsVecCalc,CtVecCalc,CkVecCalc,CBVecCalc,CNVecCalc,CTVecCalc,CjVecCalc,CaVecCalc,CvVecCalc,CurveNormalAccelerationVecCalc,CurveTangentialAccelerationVecCalc,CurveArcLengthVecCalc,CurveSpeedVecCalc,CurveTorsionVecCalc,CurveCurvatureVecCalc,CurveBinormalVecCalc,CurveNormalVecCalc,CurveTangentVecCalc,CurveJerkVecCalc,CurveAccelerationVecCalc,CurveVelocityCaNCaTCLCsCtCkCBCNCTCjCaCvCurveNormalAccelerationCurveTangentialAccelerationCurveArcLengthCurveSpeedCurveTorsionCurveCurvatureCurveBinormalCurveNormalCurveTangentCurveJerkCurveAccelerationCurveVelocityfrenetFrenetcurve"VecCalc,Angleangle"'VecCalc,&xVecCalc,Crosscross`&x`&x"4VecCalc,DetVecCalc,DeterminantdetDetdeterminant"^VecCalc,LPMDVecCalc,LeadingPrincipalMinorDeterminantsLPMDleadingprincipalminordeterminants"PVecCalc,r2dVecCalc,d2rVecCalc,rad2degVecCalc,deg2radr2dd2rrad2degdeg2rad"\VecCalc,r2pVecCalc,p2rVecCalc,rect2polarVecCalc,polar2rectr2pp2rrect2polarpolar2rect"VecCalc,c2sVecCalc,s2cVecCalc,r2sVecCalc,s2rVecCalc,r2cVecCalc,c2rVecCalc,cyl2sphVecCalc,sph2cylVecCalc,rect2sphVecCalc,sph2rectVecCalc,rect2cylVecCalc,cyl2rectc2ss2cr2ss2rr2cc2rcyl2sphsph2cylrect2sphsph2rectrect2cylcyl2rect"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Curve1Mathematics/Packages/VecCalc/Commands/CurveNormalCurve3Mathematics/Packages/VecCalc/Commands/CurveBinormalCurve4Mathematics/Packages/VecCalc/Commands/CurveCurvatureCurve2Mathematics/Packages/VecCalc/Commands/CurveTorsion CurveForget(Mathematics/Packages/VecCalc/CurveForget CurveForget1Mathematics/Packages/VecCalc/Commands/CurveForget CurveForget,Mathematics/Packages/VecCalc/Aliases/Cforget Determinant(Mathematics/Packages/VecCalc/Determinant Determinant1Mathematics/Packages/VecCalc/Commands/Determinant Determinant(Mathematics/Packages/VecCalc/Aliases/DetDiffops$Mathematics/Packages/VecCalc/Diffops Divergence'Mathematics/Packages/VecCalc/Divergence Divergence0Mathematics/Packages/VecCalc/Commands/Divergence Divergence(Mathematics/Packages/VecCalc/Aliases/DivDot Mathematics/Packages/VecCalc/DotDot)Mathematics/Packages/VecCalc/Commands/DotCurve0Mathematics/Packages/VecCalc/Commands/CurveSpeedCurve4Mathematics/Packages/VecCalc/Commands/CurveArcLengthCurveAMathematics/Packages/VecCalc/Commands/CurveTangentialAccelerationCurve=Mathematics/Packages/VecCalc/Commands/CurveNormalAccelerationCurve'Mathematics/Packages/VecCalc/Aliases/CvCurve'Mathematics/Packages/VecCalc/Aliases/CaCurve'Mathematics/Packages/VecCalc/Aliases/CjCurve'Mathematics/Packages/VecCalc/Aliases/CTCurve'Mathematics/Packages/VecCalc/Aliases/CN"%{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Cour ier" 1} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 259 23 "VecCalc[CurveForget] - " }{TEXT -1 42 "Clears Remembe r Tables from Curve Analysis" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" } {TEXT -1 48 " - The alias can be used after execution of the " } {HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 257 26 " Cforget = CurveForget" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 "\n" }{TEXT 258 71 " Cu rveForget(r,s...) Cforget(r,s...) VecCalc[CurveForget](r,s...)" }} {PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 14 " r,s... - " }{TEXT -1 93 "a sequence of curves, each in th e form of a list or Vector of expressions with one parameter." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 20 "The commands in the " }{TEXT 260 7 "VecC alc" }{TEXT -1 22 " package to perform a " }{HYPERLNK 17 "Frenet analy sis of a curve" 2 "Curve" "" }{TEXT -1 10 " (such as " }{TEXT 271 13 " CurveVelocity" }{TEXT -1 5 " and " }{TEXT 272 17 "CurveAcceleration" } {TEXT -1 113 ") use remember tables to store their results. This cuts down on computing time for other commands. The command " }{TEXT 273 14 "CurveForget(r)" }{TEXT -1 44 " clears these remember tables for th e curve " }{TEXT 277 1 "r" }{TEXT -1 15 ". The command " }{TEXT 274 19 "CurveForget(r,s...)" }{TEXT -1 49 " clears these remember tables f or all the curves " }{TEXT 275 6 "r,s..." }}{PARA 15 "" 0 "" {TEXT -1 28 "This command is part of the " }{TEXT 261 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 276 11 "CurveForget " }{TEXT -1 35 " only after performing the command " }{TEXT 263 5 "wit h(" }{TEXT 262 7 "VecCalc" }{TEXT 264 1 ")" }{TEXT -1 4 " or " }{TEXT 265 5 "with(" }{TEXT 266 7 "VecCalc" }{TEXT 267 14 ", CurveForget)" } {TEXT -1 55 ". The command can always be accessed in the long form " }{TEXT 268 7 "VecCalc" }{TEXT 269 13 "[CurveForget]" }{TEXT -1 13 ". \+ The alias " }{TEXT 270 7 "Cforget" }{TEXT -1 47 " can be used only aft er performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" } {TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCal c): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r:=;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'RTABLEG6%\" )![8H\"-%'VECTORG6#7%%\"tG,$*&\"\"#\"\"\"-%$sinG6#F-F1F1,$*&F0F1-%$cos GF4F1F1&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CB(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")gN\"H\"-%' VECTORG6#7%,$*(\"\"#\"\"\"\"\"&!\"\"F/#F.F-F0,$*&#F.F/F.*&F/F1-%$cosG6 #%\"tGF.F.F.,$*&#F.F/F.*&F/F1-%$sinGF8F.F.F0&%'VectorG6#%$rowG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cforget(r); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " Curve" 2 "Curve" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12913480 12913560 }{RTABLE M7R0 I5RTABLE_SAVE/12913480X*%)anythingG6"6"[gl!$%!!!"$"$%"tG,$-%$sinG6#F'""#,$-%$co sGF+F,6" } {RTABLE M7R0 I5RTABLE_SAVE/12913560X*%)anythingG6"6"[gl!$%!!!"$"$,$*$""&#"""""##!"#F),$*&F)F *-%$cosG6#%"tGF+#F+F),$*&F)F*-%$sinGF3F+#!""F)6" } #%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CB(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")gN\"H\"-%' VECTORG6#7%,$*(\"\"#\"\"\"\"\"&!\"\"F/#F.F-F0,$*&#F.F/F.*&F/F1-%$cosG6 #%\"tGF.F.F.,$*&#F.F/F.*&F/F1-%$sinGF8F.F.F0&%'VectorG6#%$rowG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cforget(r); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip &->"&."&@"&x"Angle"AngleConversion"CB"CL"CN"CT"Ca"CaN"CaT"Cforget"Cj"Ck"CoordConversion2D"CoordConversion3D"Cross"Cs"Ct"Curl"Curve"CurveAcceleration"CurveArcLength" CurveBinormal"CurveCurvature" CurveForget" CurveJerk" CurveNormal"CurveNormalAcceleration" CurveSpeed" CurveTangent"CurveTangentialAcceleration" CurveTorsion" CurveVelocity"Cv"D"Det" Determinant"Diff"DifferentialOperators"Diffops"Div" Divergence"Dot"EF"Frenet"Grad"Gradient"Hess"Hessian"Int"JDet"Jac"JacDet"Jacobian"JacobianDeterminant"JacobianMatrix"LPMD"Lap" Laplacian"!LeadingPrincipalMinorDeterminants"Length"Limit" LineIntScalar"LineIntScalarInert" LineIntVector"LineIntVectorInert"Lis"Liv"MF" MakeFunction"MappedFunctions"MaxMin"Muint" MultiMaxMin" Multipleint"OutputMatrixType" OutputTypes"OutputVectorType"1Mathematics/Packages/VecCalc/Commands/DivergenceDivergence*Mathematics/Packages/VecCalc/Commands/DotDot/Mathematics/Packages/VecCalc/Commands/Gradient Gradient/Mathematics/Packages/VecCalc/Commands/Hessian Hessian*Mathematics/Packages/VecCalc/Commands/IntMappedFunctions:Mathematics/Packages/VecCalc/Commands/JacobianDeterminantJacobianDeterminant5Mathematics/Packages/VecCalc/Commands/JacobianMatrixJacobianMatrix0Mathematics/Packages/VecCalc/Commands/LaplacianLaplacian5Mathematics/Packages/VecCalc/Commands/CurveArcLength Curve4Mathematics/Packages/VecCalc/Commands/CurveBinormal Curve5Mathematics/Packages/VecCalc/Commands/CurveCurvature Curve2Mathematics/Packages/VecCalc/Commands/CurveForgetCurveForget0Mathematics/Packages/VecCalc/Commands/CurveJerk Curve2Mathematics/Packages/VecCalc/Commands/CurveNormal Curve>Mathematics/Packages/VecCalc/Commands/CurveNormalAcceleration Curve1Mathematics/Packages/VecCalc/Commands/CurveSpeed Curve3Mathematics/Packages/VecCalc/Commands/CurveTangent CurveBMathematics/Packages/VecCalc/Commands/CurveTangentialAcceleration Curve3Mathematics/Packages/VecCalc/Commands/CurveTorsion Curve4Mathematics/Packages/VecCalc/Commands/CurveVelocity Curve(Mathematics/Packages/VecCalc/Commands/DMappedFunctions2Mathematics/Packages/VecCalc/Commands/DeterminantDeterminant+Mathematics/Packages/VecCalc/Commands/DiffMappedFunctionsematics/Packages/VecCalc/Commands/CurveSpeed Curve} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 48 " \+ Analysis of a Surface using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }}{PARA 256 "" 0 "" {TEXT -1 58 " c ommand(R,u,v) alias(R,u,v) VecCalc[command](R,u,v)" }}{PARA 0 "" 0 "" {TEXT  26 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT 265 15 " comm and - " }{TEXT -1 29 "a command from the list below" }}{PARA 0 "" 0 "" {TEXT 266 15 " alias - " }{TEXT -1 28 "an alias from the list below" }}{PARA 0 "" 0 "" {TEXT 256 15 " R - " }{TEXT -1 76 "a surface in the form of a list or Vector of 3 expressions with 2 par ameters" }}{PARA 0 "" 0 "" {TEXT 261 15 " " }{TEXT -1 42 "or an expression which simplifies to such." }}{PARA 0 "" 0 "" {TEXT 260 15 " u, v - " }{TEXT -1 26 "parameters for the surfac e" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 " " 0 "" {TEXT -1 140 "These commands are designed to perform an analysi s of a surface. Each command has a shorter alias. The commands only \+ work in 3 dimensions." }}{PARA 15 "" 0 "" {TEXT -1 95 "Below is a list of each command, the corresponding alias and a short description of i ts action." }}{PARA 0 "" 0 "" {TEXT 257 34 " SurfaceTangents ST - " }{TEXT -1 29 "Calculate the Tan gent Vectors" }}{PARA 0 "" 0 "" {TEXT 258 34 " SurfaceNormal SN - " }{TEXT -1 27 "Cal culate the Normal Vector" }}{PARA 0 "" 0 "" {TEXT 267 34 " SurfaceNor malLength SNL - " }{TEXT -1 35 "Calculate the Length of the Norm al " }}{PARA 0 "" 0 "" {TEXT 268 34 " SurfaceArea SA - " }{TEXT -1 26 "Calculate the Surface Area" }}{PARA 0 "" 0 "" {TEXT 259 34 " SurfaceForget Sforget - " }{TEXT -1 49 "Clear the re member tables for the above commands " }}{PARA 15 "" 0 "" {TEXT -1 48 "To use the command name, you must first execute " }{TEXT 262 13 "with (VecCalc)" }{TEXT -1 4 " or " }{TEXT 263 21 "with(VecCalc,command)" } {TEXT -1 58 ". To use the aliases, you must first execute the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}{PARA 15 " " 0 "" {TEXT -1 195 "The surface must be in the form of a list or Vect or of 3 expressions with 2 parameters, except that Maple will first at tempt to simplify it to this form. Such a surface may be plotted usin g the " }{HYPERLNK 17 "plot3d" 2 "plot3d" "" }{TEXT -1 36 " command wi th a parametric argument." }}{PARA 15 "" 0 "" {TEXT -1 20 "Each comman d uses a " }{HYPERLNK 17 "remember" 2 "remember" "" }{TEXT -1 165 " ta ble to speed up the computation. These tables may be cleared after fin ishing the analysis of a surface to avoid cluttering the memory. This is done by using the " }{HYPERLNK 17 "SurfaceForget" 2 "CurveForget" "" }{TEXT -1 18 " command from the " }{TEXT 269 7 "VecCalc" }{TEXT -1 22 " package or its alias " }{TEXT 264 7 "Sforget" }{TEXT -1 1 "." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Examples: " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")%yXP\"-%'VECTORG6#7$,$*&,&-%$cosG 6#%\"tG!\"\"*&F1\"\"\"-%$sinGF0F4F4F4,&*$)F1\"\"#F4F4F4F4#F2F:F2*&F7F; ,&F5F4*&F1F4F.F4F4F4&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A 3D Example using aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R:=;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'RTABLEG6%\")Ceu8-%'MATRIXG6#7%7#*&%\"rG\"\"\"- %$cosG6#%\"tGF07#*&F/F0-%$sinGF3F07#F4&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot3d(R, r=0..3, t=0..6*Pi, grid=[5,73]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Rr,Rt:=ST (R,r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#RrG%#RtG6$-%'RTABLEG6 %\")keu8-%'MATRIXG6#7%7#-%$cosG6#%\"tG7#-%$sinGF37#\"\"!&%'VectorG6#%' columnG-F)6%\")/fu8-F-6#7%7#,$*&%\"rG\"\"\"F6FH!\"\"7#*&FGFHF1FH7#FHF: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "N:=SN(R,r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'RTABLEG6%\")Wfu8-%'MATRIXG6#7%7#-%$ sinG6#%\"tG7#,$-%$cosGF0!\"\"7#%\"rG&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "lenN:=SNL(R,r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lenNG*$,&\"\"\"F'*$)%\"rG\"\"#F'F'#F'F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=SA(R,r,t); A(0,3,0,6*Pi); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6&%\"aG%\"bG%\"c G%\"dG6\"6$%)operatorG%&arrowGF+-%$IntG6$-F06$*$,&\"\"\"F6*$)%\"rG\"\" #F6F6#F6F:/F9;9$9%/%\"tG;9&9'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$IntG6$-F$6$*$,&\"\"\"F**$)%\"rG\"\"#F*F*#F*F./F-;\"\"!\"\"$/%\"tG; F2,$*&\"\"'F*%#PiGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"*\" \"\"\"#5#F&\"\"#%#PiGF&F&*(\"\"$F&-%#lnG6#,&F,!\"\"*$F'F(F&F&F*F&F1" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Sforget(R);" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " SurfaceForget" 2 "SurfaceForget" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Cu rve" 2 "Curve" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13745784 13745824 13745864 13745904 13745944 }{RTABLE M7R0 I5RTABLE_SAVE/13745784X*%)anythingG6"6"[gl!$%!!!"#"#,$*&,&-%$cosG6#%"tG!""*&F-" ""-%$sinGF,F0F0F0,&*$F-""#F0F0F0#F.F5F.*&F3F6,&F1F0*&F-F0F*F0F0F06" } {RTABLE M7R0 I5RTABLE_SAVE/13745824X*%)anythingG6"6"[gl!#%!!!"$"$*&%"rG"""-%$cosG6#%"tGF)*&F (F)-%$sinGF,F)F-6" } {RTABLE M7R0 I5RTABLE_SAVE/13745864X*%)anythingG6"6"[gl!#%!!!"$"$-%$cosG6#%"tG-%$sinGF)""!6" } {RTABLE M7R0 I5RTABLE_SAVE/13745904X*%)anythingG6"6"[gl!#%!!!"$"$,$*&%"rG"""-%$sinG6#%"tGF*! ""*&F)F*-%$cosGF-F*F*6" } {RTABLE M7R0 I5RTABLE_SAVE/13745944X*%)anythingG6"6"[gl!#%!!!"$"$-%$sinG6#%"tG,$-%$cosGF)!"" %"rG6" } f Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " SurfaceForget" 2 "SurfaceForget" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Cu rve" 2 "Curve" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13745784 13745824 13745864 13745904 13745944 }{RTABLE M7R0 I5RTABLE_SAVE/13745784X*%)anythingG6"6"[gl!$%!!!"#"#beginn"bel"belmont$""""""""""""""""""""""""""""""""""below """""ber"best ""between"bgC""""""""""""""""bgf+"""""""""" bianmatrix"binormal"ble ""bleg"bles ""blish"same""""""save;"""""""""""""""sca"scal"scala"scalar#"""""""" scalarinert"scalarpo" scalarpot "" scalarpoten"scalarpotential"""""""script"se"""""second """""""sed"""select ""separate"seq"sequ"sequenc0"""""""""""""""""""""""""""""""""set """"sets"sever"several"""sforget """"let"letter"lf"lfgf"lfunct"lgebra"lgf"li """"""lias""""""libnam"like"lim"limit ""lin ""linal ""linalg7>"""""""""""""line """"linear"lineara" linearalgeb" linearalgebr" linearalgebra+""""""""""lineg"linei"""linein"lineint"""inggglibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaltimesbulletitemanalysisurfacusingveccalcpackagcallsequencommandaliacommandparametercommlistbelowformvectorexpresswithparametersimplifsuchdescriptthesdesignperformeachshortonlyworkdimenscorrespondtsactionsurfacetangentstcalculattangsurfacenormalsncalculatsurfacenormallengthsnllengthnormalsurfaceareasaareasurfaceforgetsforgetclearremembtablabovusenameyoumustfirstexecutaliasvcaliavectexceptwillattemptmayplottusinplotwithparametricargumcommanusesremembtablespeedupcomputatafterfinishingavoidcluttermemordonecurveforgetexamplrtablegyxpvectorgcosgtgsingfrowgcossinrgceumatrixgtgfcolumngpigridrrrtrrgrtgkeufufhfgfhffhfngwfusingcosgflennlenngvaluagfagbgdgoperatorgarrowgfintgpigflngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvecurvrtablsaveanyth"#{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYL"w.{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 259 25 "VecCalc[SurfaceForget] - " }{TEXT -1 44 "Clears Remem ber Tables from Surface Analysis" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias: " }{TEXT -1 48 " - The alias can be used after execution of the " } {HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 257 28 " Sforget = SurfaceForget" }}{PARA 0 "" 0 " " {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 "\n" }{TEXT 258 75 " \+ SurfaceForget(R,S...) Cforget(R,S...) VecCalc[SurfaceForget](R,S.. .)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 14 " R,S... - " }{TEXT -1 96 "a sequence of Surfaces, ea ch in the form of a list or Vector of 3 expressions with 2 parameters. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " \+ " }}{PARA 15 "" 0 "" {TEXT -1 20 "The commands in the " }{TEXT 260 7 " VecCalc" }{TEXT -1 23 " package to perform an " }{HYPERLNK 17 "Analysi s of a Surface" 2 "Surface" "" }{TEXT -1 10 " (such as " }{TEXT 271 15 "SurfaceTangents" }{TEXT -1 5 " and " }{TEXT 272 13 "SurfaceNormal " }{TEXT -1 113 ") use remember tables to store their results. This c uts down on computing time for other commands. The command " }{TEXT 273 16 "SurfaceForget(R)" }{TEXT -1 62 " clears these remember tables \+ for the Surface R. The command " }{TEXT 274 21 "SurfaceForget(R,S...) " }{TEXT -1 51 " clears these remember tables for all the Surfaces " } {TEXT 276 6 "R,S..." }}{PARA 15 "" 0 "" {TEXT -1 28 "This command is p art of the " }{TEXT 261 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 275 13 "SurfaceForget" }{TEXT -1 35 " on ly after performing the command " }{TEXT 263 5 "with(" }{TEXT 262 7 "V ecCalc" }{TEXT 264 1 ")" }{TEXT -1 4 " or " }{TEXT 265 5 "with(" } {TEXT 266 7 "VecCalc" }{TEXT 267 16 ", SurfaceForget)" }{TEXT -1 55 ". The command can always be accessed in the long form " }{TEXT 268 7 " VecCalc" }{TEXT 269 15 "[SurfaceForget]" }{TEXT -1 13 ". The alias " }{TEXT 270 7 "Sforget" }{TEXT -1 47 " can be used only after performin g the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R:=;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'RTABLEG6%\"))oxL\"-%'VECTORG 6#7%%\"tG*&%\"rG\"\"\"-%$sinG6#F-F0*&F/F0-%$cosGF3F0&%'VectorG6#%$rowG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "SN(R,r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")oxP8-%'VECTORG6#7%,$%\"rG!\"\"-% $cosG6#%\"tG,$-%$sinGF0F-&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Sforget(R); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "V ecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface" 2 "Surface" "" } {TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13377688 13377768 }{RTABLE M7R0 I5RTABLE_SAVE/13377688X*%)anythingG6"6"[gl!$%!!!"$"$%"tG*&%"rG"""-%$sinG6#F'F** &F)F*-%$cosGF-F*6" } {RTABLE M7R0 I5RTABLE_SAVE/13377768X*%)anythingG6"6"[gl!$%!!!"$"$,$%"rG!""-%$cosG6#%"tG,$-%$ sinGF,F)6" } EG6%\"))oxL\"-%'VECTORG 6#7%%\"tG*&%\"rG\"\"\"-%$sinG6#F-F0*&F/F0-%$cosGF3F0&%'VectorG6#%$rowG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "SN(R,r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")oxP8-%'VECTORG6#7%,$%\"rG!\"\"-% $cosG6#%\"tG,$-%$sinGF0F-&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Sforget(R); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 VecCalc,Limit"VecCalc,LineIntScalar"VecCalc,LineIntScalarInert"VecCalc,LineIntVector"VecCalc,LineIntVectorInert" VecCalc,Lis" VecCalc,Liv" VecCalc,MF"VecCalc,MakeFunction" VecCalc,Muint"VecCalc,Multipleint"VecCalc,OutputMatrixType"VecCalc,OutputVectorType" VecCalc,SA" VecCalc,SN" VecCalc,SNL" VecCalc,SPot" VecCalc,ST"VecCalc,ScalarPotential"VecCalc,Sforget" VecCalc,Sis" VecCalc,Siv"VecCalc,SurfaceArea"VecCalc,SurfaceForget"VecCalc,SurfaceIntScalar"VecCalc,SurfaceIntScalarInert"VecCalc,SurfaceIntVector"VecCalc,SurfaceIntVectorInert"VecCalc,SurfaceNormal"VecCalc,SurfaceNormalLength"VecCalc,SurfaceTangents"VecCalc,VCalias" VecCalc,VPot"outputmatrixtyp""""outputmatrixtypeg" outputtyp"outputv" outputvec" outputvectort ""outputvectortyp""""outputvectortypeg"outtyp+"" """""ov"""ove"over "" overridden"ovl"own"oxl"oxp"pa"""pac""""packa"packagO"""""""""""""""""""""""""""""""""""98{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" - 1 269 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Times " 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Tim es" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "Times" 1 12 0 0 0 !0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Tim es" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 294 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 295 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "Tim es" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }"{CSTYLE "" -1 302 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 303 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 312 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 313 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "Courier" 1 #10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 321 " " 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 322 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 " " 1 $10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 339 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 340 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Norm al" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Fixed Width" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 49 " \+ Differential Operators using the VecCalc Package" }}{PARA 0 "" 0 "" {TEXT 26 18 "Callin%g Sequences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 64 " MakeFunction(in,out) VecCalc[MakeFunction]( in,out)" }}{PARA 257 "" 0 "" {TEXT -1 67 " evalFunction(fnc,point) \+ VecCalc[evalFunction](fnc,point)" }}{PARA 257 "" 0 "" {TEXT -1 68 " Gradient(f,vars,outtype) VecCalc[Gradient](f,vars,outtyp e)" }}{PARA 257 "" 0 "" {TEXT -1 67 " Hessian(f,vars,outtype) \+ VecCalc[Hessian](f,vars,outtype)" }}{PARA 257 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 305 36 "LeadingPrincipalMinorDeterminants(M)" }}{PARA 257 " " 0 "" {TEXT -1 43 " VecCalc[" } {TEXT 306 37 "LeadingPrincipalMinorDeterminants](M)" }}{PARA 257 "" 0 "" {TEXT -1 74 " JacobianMatrix(T,vars,outtype) VecCalc[JacobianMat rix](T,vars,outtype)" }}{PARA 257 "" 0 "" {TEXT -1 71 " JacobianDete rminant(T,vars) VecCalc[JacobianDeterminant](T,vars)" }}{PARA 257 "" 0 "" {TEXT -1 62 " Divergence(F,vars) VecCalc[Diverg ence](F,vars)" }}{PARA 257 "" 0 "" {TEXT -1 6&4 " Curl(F,vars,outtype ) VecCalc[Curl](F,vars,outtype)" }}{PARA 257 "" 0 "" {TEXT -1 61 " Laplacian(fnc,vars) VecCalc[Laplacian](f,vars)" }}{PARA 257 "" 0 "" {TEXT -1 67 " ScalarPotential(F,vars) Ve cCalc[ScalarPotential](F,vars)" }}{PARA 257 "" 0 "" {TEXT -1 67 " Ve ctorPotential(F,vars,outtype) VecCalc[VectorPotential](F,vars)" }} {PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 13 " in - " }{TEXT -1 77 "a name or a list or Vector of nam es representing the independent variable(s)." }}{PARA 0 "" 0 "" {TEXT 257 13 " out - " }{TEXT -1 134 "an expression or list, Vector, M atrix or Array of expressions representing the scalar, vector, matrix \+ or array value of the function,\n" }{TEXT 307 13 " " } {TEXT -1 42 "OR nested lists and Arrays of expressions." }}{PARA 0 "" 0 "" {TEXT 261 13 " fnc - " }{TEXT -1 70 "a function to be evalu ated or differentiated, in the form produced by " }{TEXT 309 12 "MakeF unction" '}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 310 13 " point - \+ " }{TEXT -1 133 "an expression or expression sequence, list or Vector \+ of expressions representing the point at which the function should be \+ evaluated." }}{PARA 0 "" 0 "" {TEXT 308 13 " f - " }{TEXT -1 71 "a scalar field in the form of an arrow-defined function of n varia bles." }}{PARA 0 "" 0 "" {TEXT 262 13 " F - " }{TEXT -1 91 "a \+ vector field in the form of a list or Vector of n arrow-defined functi ons of n variables." }}{PARA 0 "" 0 "" {TEXT 263 13 " " } {TEXT -1 5 "(For " }{TEXT 311 4 "Curl" }{TEXT -1 5 " and " }{TEXT 312 15 "VectorPotential" }{TEXT -1 15 ", n must be 3.)" }}{PARA 0 "" 0 "" {TEXT 260 13 " M - " }{TEXT -1 48 "a square Matrix or list of \+ lists of expressions." }}{PARA 0 "" 0 "" {TEXT 259 13 " T - " }{TEXT -1 104 "a coordinate transformation in the form of a list or Ve ctor of n arrow-defined functions of k variables." }}{PARA 0 "" 0 "" {TEXT 264 13 " "( }{TEXT -1 5 "(For " }{TEXT 313 19 "Jacobi anDeterminant" }{TEXT -1 41 ", T must be square, i.e. k must equal n.) " }}{PARA 0 "" 0 "" {TEXT 258 13 " vars - " }{TEXT -1 1 "(" } {TEXT 314 8 "optional" }{TEXT -1 89 ") sequence, list or Vector of nam es to be used as independent variables for the function." }}{PARA 0 " " 0 "" {TEXT 315 13 " outtype - " }{TEXT 316 31 "(optional) type for the output." }}{PARA 0 "" 0 "" {TEXT 341 13 " " }{TEXT 340 4 "For " }{TEXT 329 8 "Gradient" }{TEXT -1 2 ", " }{TEXT 330 4 "Cu rl" }{TEXT -1 5 " and " }{TEXT 331 15 "VectorPotential" }{TEXT -1 19 " , the choices are: " }{TEXT 317 31 "'list', 'Vector', 'Vector[row]'" } {TEXT 327 4 " or " }{TEXT 328 17 "'Vector[column]'." }}{PARA 0 "" 0 " " {TEXT 318 13 " " }{TEXT 319 4 "For " }{TEXT 320 7 "Hessi an" }{TEXT 321 5 " and " }{TEXT 322 14 "JacobianMatrix" }{TEXT 323 19 ", the choices are: " }{TEXT 324 10 "'listlist'" }{TEXT 325 4 " or " } {TEXT 326 8 "'Matrix'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT) 332 13 " " }{TEXT 333 132 "If outtype is not specified, the type \+ of the output is determined by the type of the input or by the values \+ of the global variables " }{TEXT -1 1 " " }{HYPERLNK 17 "OutputVectorT ype" 2 "OutputVectorType" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "Output MatrixType" 2 "OutputMatrixType" "" }{TEXT -1 18 " which default to " }{TEXT 334 13 "'Vector[row]'" }{TEXT 335 0 "" }{TEXT 336 3 " or" } {TEXT -1 1 " " }{TEXT 337 8 "'Matrix'" }{TEXT 338 0 "" }{TEXT 339 1 ". " }{TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description: " }}{PARA 15 "" 0 "" {TEXT -1 285 "These commands are designed to help with the definition, differentiation and anti-differentiation of scal ar fields, vector fields and coordinate transformations (including par ametrized surfaces). Most commands have a shorter alias. Most comman ds work in any dimension. However, the " }{TEXT 301 4 "Curl" }{TEXT -1 5 " and " }{TEXT 302 15 "VectorPotential" }{TEXT -1 40 " commands o nly work in three d*imensions." }}{PARA 15 "" 0 "" {TEXT 26 21 "Functio ns and Aliases" }{TEXT -1 161 ": Below is a list of each command, the \+ corresponding alias and a short description of its action. Each comman d name is linked to its own help page with examples." }}{PARA 256 "" 0 "" {TEXT 283 3 " " }{HYPERLNK 17 "MakeFunction" 2 "MakeFunction" " " }{TEXT 284 15 " MF - " }{TEXT 285 30 "Make an arrow-defined function" }}{PARA 256 "" 0 "" {TEXT 267 3 " " }{HYPERLNK 17 "evalFu nction" 2 "evalFunction" "" }{TEXT 287 15 " EF - " }{TEXT 288 34 "Evaluate an arrow-defined function" }}{PARA 256 "" 0 "" {TEXT 286 3 " " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT 268 19 " \+ Grad - " }{TEXT 269 22 "Calculate the Gradient" }}{PARA 256 "" 0 "" {TEXT 289 3 " " }{HYPERLNK 17 "Hessian" 2 "Hessian" "" } {TEXT 290 20 " Hess - " }{TEXT 291 21 "Calculate the Hessi an" }}{PARA 256 "" 0 "" {TEXT 265 3 " " }{HYPERLNK 17 "LeadingPrinci palMinorDeterminants" 2 "LeadingPrincipalMinorDet+erminants" "" }} {PARA 256 "" 0 "" {TEXT 266 30 " LPMD - " } {TEXT 270 50 "Calculate the Leading Principal Minor Determinants" }} {PARA 256 "" 0 "" {TEXT 271 3 " " }{HYPERLNK 17 "JacobianMatrix" 2 " JacobianMatrix" "" }{TEXT 272 13 " Jac - " }{TEXT 273 29 "Calcul ate the Jacobian Matrix" }}{PARA 256 "" 0 "" {TEXT 274 3 " " } {HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT 275 8 " JDet - " }{TEXT 276 34 "Calculate the Jacobian Determinant" }} {PARA 256 "" 0 "" {TEXT 292 3 " " }{HYPERLNK 17 "Divergence" 2 "Dive rgence" "" }{TEXT 293 17 " Div - " }{TEXT 294 24 "Calculate \+ the Divergence" }}{PARA 256 "" 0 "" {TEXT 295 3 " " }{HYPERLNK 17 "C url" 2 "Curl" "" }{TEXT 296 23 " - " }{TEXT 297 18 "Calculate the Curl" }}{PARA 256 "" 0 "" {TEXT 298 3 " " } {HYPERLNK 17 "Laplacian" 2 "Laplacian" "" }{TEXT 299 18 " La p - " }{TEXT 300 23 "Calculate the Laplacian" }}{PARA 256 "" 0 "" {TEXT 277 3 " " }{HYPERLNK 17 ",ScalarPotential" 2 "ScalarPotential" "" }{TEXT 278 12 " SPot - " }{TEXT 279 36 "Find a Scalar Potential if it Exists" }}{PARA 256 "" 0 "" {TEXT 280 3 " " }{HYPERLNK 17 "Ve ctorPotential" 2 "VectorPotential" "" }{TEXT 281 12 " VPot - " } {TEXT 282 36 "Find a Vector Potential if it Exists" }}{PARA 15 "" 0 " " {TEXT -1 49 "To use the command names, you must first execute " } {TEXT 303 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 304 21 "with(Ve cCalc,command)" }{TEXT -1 57 ".\nTo use the aliases, you must first ex ecute the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematic s, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "S urface" 2 "Surface" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin"- 2 "MultiMaxMin" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } }}{PARA 256 "" 0 "" {TEXT 280 3 " " }{HYPERLNK 17 "Ve ctorPotential" 2 "VectorPotential" "" }{TEXT 281 12 " VPot - " } {TEXT 282 36 "Find a Vector Potential if it Exists" }}{PARA 15 "" 0 " " {TEXT -1 49 "To use the command names, you must first execute " } {TEXT 303 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 304 21 "with(Ve cCalc,command)" }{TEXT -1 57 ".\nTo use the aliases, you must first ex ecute the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematic s, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "S urface" 2 "Surface" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin"-cag"cal"""calc"""calclab"calcul ""calcula"calculatc6"""""""" """"""""""""""""calculu ""calia"call!"""""""""""""""""""""""""""""""""can"""""""""""""""""""""""""""""""""cang"cannot""""" cartesian"cat"""catg"caut"""cb""""cbg"ccalc """""""ccess"ccosg"ce" ceintscalar" ceintvector"cess"ceu"cf"cfg"cforget """""cforgetg"cg3""""""""""""cgf#""""""""ch"chad"chang"choic"chosen""""chri"cian"cifi"cj""""cjg"ck"""ckag ""ckg"cl"""classif"clear """"clg"click"div""""dive"diver"diverg"""divergen" divergenc#""""""""divg"divid"do"""""does"""domain""""done""""dot"" """dotprod" dotproduct"doublein" doubleint"down ""drical"ds"dth"duce"duct"ea"each#""""""""eal"eccalc"""""ecpot"ect ""ector"ectorg "" ectorinert"ectorpotential"ectortyp"ecut"ed"""""asbeloweachcorrespondactionlinkownpageexamplmfmakeevalfunctionefgradcalculathessleadingprincipalminordeterminantlpmdleadprincipalminordeterminantjaccalculatejacobianjdetdivergencdivurllaspotfindpotentialexistvpotuseyoufirstexecutntoexecutvcaliacopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvurfacmultimaxminorpotentialvectorpotentialparameternamelistvectornamrepresentindependvariablexpressatrixarrascalarmatrixvalufunctnestarrayevaluateddifferentiatformproducmakefunctatevaluatfieldarrowdefinvariablesfunctionsmustsquarcoordinattransformatctorfunctionjacobiandeterminantequaloptionalusedtypeoutputcurlchoicrowcolumnhessilistlistspecifidetermininputglobaloutputvectortypeoutputvectortypmatrixtypoutputmatrixtypdefaultdescriptthescommanddesignwithdefinitantiscalarincludparametrizsurfacmosthaveshortaliacommandsworkanydimenshowevnlyfunctionsali1"dr{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 278 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 279 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 " Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Times4" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 288 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "Courier" 1 10 0 0 0 0 0 0 50 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 305 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 309 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "Tim es" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }61 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 57 " \+ Multivariable Max-Min Problems using the VecCalc Package" }}{PARA 0 " " 0 "" {TEXT 26 9 "Functions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT 276 3 " " }{HYPERLNK 17 "VecCalc[Gradient]" 2 "vec_calc[Gradie nt]" "" }{TEXT 277 7 " - " }{TEXT 278 22 "Calculate the Gradient" }}{PARA 256 "" 0 "" {TEXT 258 3 " " }{HYPERLNK 17 "student[equate]" 2 "vec_calc[GRAD]" "" }{TEXT 259 9 " - " }{TEXT 279 74 "Set the \+ Gradient equ7al to zero or set up the Lagrange Multiplier equations" }} {PARA 256 "" 0 "" {TEXT 260 3 " " }{HYPERLNK 17 "solve" 2 "solve" " " }{TEXT 261 19 " - " }{TEXT 280 31 "Solve for exact c ritical points" }}{PARA 256 "" 0 "" {TEXT 262 3 " " }{HYPERLNK 17 "f solve" 2 "fsolve" "" }{TEXT 263 18 " - " }{TEXT 281 45 "Solve for approximate decimal critical points" }}{PARA 256 "" 0 "" {TEXT 264 3 " " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT 265 15 " - " }{TEXT 282 33 "Evaluate solutions which conta in " }{HYPERLNK 17 "RootOf" 2 "RootOf" "" }{TEXT 283 2 "'s" }}{PARA 256 "" 0 "" {TEXT 266 3 " " }{HYPERLNK 17 "VecCalc[Hessian]" 2 "VecC alc[Hessian]" "" }{TEXT 267 8 " - " }{TEXT 284 21 "Calculate the \+ Hessian" }}{PARA 256 "" 0 "" {TEXT 268 3 " " }{HYPERLNK 17 "VecCalc[ LeadingPrincipalMinorDeterminants]" 2 "VecCalc[LeadingPrincipalMinorDe terminants]" "" }{TEXT 269 5 " - " }{TEXT 285 86 "Calculate the Lead ing Principal Minor Determinants to apply t8he Second Derivative Test" }}{PARA 256 "" 0 "" {TEXT 270 3 " " }{HYPERLNK 17 "subs" 2 "subs" " " }{TEXT 271 20 " - " }{TEXT 286 49 "Convert a soluti on set into a list of coordinates" }}{PARA 256 "" 0 "" {TEXT 274 3 " \+ " }{HYPERLNK 17 "VecCalc[MakeFunction]" 2 "VecCalc[MakeFunction]" "" }{TEXT 275 3 " - " }{TEXT 288 30 "Make an arrow-defined function" }} {PARA 256 "" 0 "" {TEXT 303 3 " " }{HYPERLNK 17 "VecCalc[&->]" 2 "Ve cCalc[MakeFunction]" "" }{TEXT 304 12 " - " }{TEXT 305 30 "Ma ke an arrow-defined function" }}{PARA 256 "" 0 "" {TEXT 300 3 " " } {HYPERLNK 17 "VecCalc[evalFunction]" 2 "VecCalc[evalFunction]" "" } {TEXT 301 3 " - " }{TEXT 302 34 "Evaluate an arrow-defined function" } }{PARA 256 "" 0 "" {TEXT 309 3 " " }{HYPERLNK 17 "VecCalc[&@]" 2 "Ve cCalc[evalFunction]" "" }{TEXT 310 13 " - " }{TEXT 312 34 "E valuate an arrow-defined function" }}{PARA 0 "" 0 "" {TEXT 26 8 "Alias es:" }{TEXT -1 52 " - These aliases can be used after execution of the " 9}{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT 256 20 " Grad = Gradient" }}{PARA 0 "" 0 "" {TEXT 289 19 " Hess = Hessian" }}{PARA 0 "" 0 "" {TEXT 290 45 " \+ LPMD = LeadingPrincipalMinorDeterminants" }}{PARA 0 "" 0 "" {TEXT 257 24 " MF = makeFunction" }}{PARA 0 "" 0 "" {TEXT 313 24 " M F = evalFunction" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descript ion:" }}{PARA 15 "" 0 "" {TEXT -1 147 "These commands are designed to \+ help with multidimensional max-min problems. Below are examples of bo th the unconstrained and constrained problems." }}{PARA 15 "" 0 "" {TEXT -1 22 "To use the functions, " }{TEXT 291 8 "Gradient" }{TEXT -1 2 ", " }{TEXT 292 7 "Hessian" }{TEXT -1 2 ", " }{TEXT 293 33 "Leadi ngPrincipalMinorDeterminants" }{TEXT -1 2 ", " }{TEXT 294 12 "MakeFunc tion" }{TEXT -1 6 ", and " }{TEXT 314 12 "evalFunction" }{TEXT -1 20 " and the operators, " }{TEXT 315 3 "&->" }{TEXT -1 5 " and " }{TEXT 316 2 "&@" }{TEXT -1 25 ", you must :first execute " }{TEXT 295 13 "wit h(VecCalc)" }{TEXT -1 4 " or " }{TEXT 296 107 "with(VecCalc,Gradient, \+ Hessian, LeadingPrincipalMinorDeterminants, MakeFunction, evalFunction , `&->`, `&@`)" }{TEXT -1 28 ". To use the command name, " }{TEXT 297 6 "equate" }{TEXT -1 25 ", you must first execute " }{TEXT 298 13 "with(student)" }{TEXT -1 4 " or " }{TEXT 299 21 "with(student, equate )" }{TEXT -1 58 ". To use the aliases, you must first execute the com mand " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Examples: " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 "The Unconstrained Problem:" }}{PARA 0 "" 0 "" {TEXT -1 111 "Find all critical points of a function and classify each as a local maximum, a local minimum or a saddle point." }}{PARA 5 "" 0 "" {TEXT -1 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 46 "Classify the critical points of the function: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT; 1 0 32 "f:=(x,y)->x*y*exp(-x^2/2-y^2/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*(9$\"\" \"9%F/-%$expG6#,&*&#F/\"\"#F/*$)F.F7F/F/!\"\"*&#F/\"\")F/*$)F0F7F/F/F: F/F)F)F)" }}}{PARA 5 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 " " {TEXT -1 83 "Compute the gradient of f, set it equal to zero and sol ve for the critical points: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=Grad(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delfG-%'RTABL EG6%\")w#oV\"-%'VECTORG6#7$f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF1,$* (9%\"\"\"-%$expG6#,&*&#F8\"\"#F8*$)9$F?F8F8!\"\"*&#F8\"\")F8*$)F7F?F8F 8FCF8,&F8FCF@F8F8FCF1F1F1f*F.F1F2F1,$*&#F8\"\"%F8*(FBF8F9F8,&FNFCFGF8F 8F8FCF1F1F1&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqs:=equate(delf&@(x,y),[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,$*&#\"\"\"\"\"%F**(%\"xGF*-%$expG6#,&*&\"\"#!\"\"F-F3F4* &\"\")F4%\"yGF3F4F*,&F+F4*$)F7F3F*F*F*F*F4\"\"!/,$*(F7F*F.F*,&F*F4*$)F -F3F*F*F*F4F;" }}}{EXCHG {PARA 0 ">< " 0 "" {MPLTEXT 1 0 26 "critpts:=s olve(eqs,\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(critptsG6'<$/ %\"xG\"\"!/%\"yGF)<$/F(\"\"\"/F+\"\"#<$F-/F+!\"#<$/F(!\"\"F/<$F5F2" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p1:=subs(critpts[1],[x,y]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p2:=subs(critpts[2],[x,y]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#p2G7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "p3:=subs(critpts[3],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G7$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p4:=subs(critpts[4],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G7$!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p5:=subs(critpts[5],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G7$!\"\"!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 210 "Use the second derivative test to determine if each critical point is a m aximum, a minimum or a saddle point. Note, the test may fail. First, compute the Hessia=n and the leading principal minor determinants: " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Hf:=Hess(f)&@(x,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HfG-%'RTABLEG6%\")CV$H\"-%'MATRIXG6 #7$7$**%\"xG\"\"\"%\"yGF0-%$expG6#,&*&\"\"#!\"\"F/F7F8*&\"\")F8F1F7F8F 0,&\"\"$F8*$)F/F7F0F0F0,$*&#F0\"\"%F0*&F2F0,*FBF0*&FBF0F>F0F8*$)F1F7F0 F8*&F>F0FGF0F0F0F0F07$F?,$*&#F0\"#;F0**F/F0F1F0F2F0,&\"#7F8FFF0F0F0F0% 'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "LPMD(Hf); simpl ify([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$**%\"xG\"\"\"%\"yGF%-%$ex pG6#,&*&\"\"#!\"\"F$F,F-*&\"\")F-F&F,F-F%,&\"\"$F-*$)F$F,F%F%F%,2*&#\" \"&\"\"%F%*(F3F%)F&F,F%)F'F,F%F%F%*&#F%\"#;F%*(F3F%)F&F8F%F;F%F%F-*&#F %F8F%*()F$F8F%F:F%F;F%F%F-*$F;F%F-*(F,F%F;F%F3F%F%*&#F%F,F%*&F;F%F:F%F %F%*&F;F%FDF%F-*&#F%F>F%*&F;F%F@F%F%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$**%\"xG\"\"\"%\"yGF&-%$expG6#,&*&\"\"#!\"\"F%F-F.*&\"\")F.F'F- F.F&,&\"\"$F.*$)F%F-F&F&F&,$*&#F&\"#;F&*&-F)6#,&F3F.*&\"\"%F.F'F-F.F&, 2*(\"#?F&F4F&)F'F-F&F.*&)F'F>F&F4F&F&*(F>F&)F%F>F&F>BF&F&F8F&*&\"#KF&F4 F&F.*&F0F&FBF&F.*&F8F&FFF&F&*$FDF&F&F&F&F." }}}{PARA 0 "" 0 "" {TEXT -1 116 "At each critical point, evaluate the Hessian and the leading p rincipal minor determinants and interpret the results:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "'p1'=p1; H1:=Hf&@p1; D1,D2:=LPM D(H1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p1G7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G-%'RTABLEG6%\")KrF9-%'MATRIXG6#7$7$\"\"! \"\"\"7$F/F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#D1G%#D2G 6$\"\"!!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 71 "Since D2 is negative, th e first critical point (0,0) is a saddle point." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "'p2'=p2; H2:=Hf&@p2; D1,D2:=LPMD(H2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p2G7$\"\"\"\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#H2G-%'RTABLEG6%\")CS`9-%'MATRIXG6#7$7$,$*&\"\"%\" \"\"-%$expG6#!\"\"F1F5\"\"!7$F6,$F2F5%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#D1G%#D2G6$,$*&\"\"%\"\"\"-%$expG6#!\"\"F+F/,$*&F*F +)F,\"\"#F+F+" }}?}{PARA 0 "" 0 "" {TEXT -1 92 "Since D2 is positive an d D1 is negative, the second critical point (1,2) is a local maximum. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "'p3'=p3; H3:=Hf&@p3; \+ D1,D2:=LPMD(H3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p3G7$\"\"\"!\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G-%'RTABLEG6%\")cj4:-%'MATRI XG6#7$7$,$*&\"\"%\"\"\"-%$expG6#!\"\"F1F1\"\"!7$F6F2%'MatrixG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#D1G%#D2G6$,$*&\"\"%\"\"\"-%$expG6 #!\"\"F+F+,$*&F*F+)F,\"\"#F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 92 "Since \+ D2 is positive and D1 is positive, the third critical point (1,-2) is \+ a local minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "'p4'=p4; H4:=Hf&@p4; D1,D2:=LPMD(H4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %#p4G7$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H4G-%'RTABLEG6 %\"(3PZ#-%'MATRIXG6#7$7$,$*&\"\"%\"\"\"-%$expG6#!\"\"F1F1\"\"!7$F6F2%' MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%#D1G%#D2G6$,$*&\"\"%\" \"\"-%$expG6#!\"\"F+F+,$*&F*F+)F,\"\"#F+F+" @}}}{PARA 0 "" 0 "" {TEXT -1 93 "Since D2 is positive and D1 is positive, the fourth critical po int (-1,2) is a local minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "'p5'=p5; H5:=Hf&@p5; D1,D2:=LPMD(H5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p5G7$!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #H5G-%'RTABLEG6%\")c\"za\"-%'MATRIXG6#7$7$,$*&\"\"%\"\"\"-%$expG6#!\" \"F1F5\"\"!7$F6,$F2F5%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$% #D1G%#D2G6$,$*&\"\"%\"\"\"-%$expG6#!\"\"F+F/,$*&F*F+)F,\"\"#F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 93 "Since D2 is positive and D1 is negative, \+ the fifth critical point (-1,-2) is a local maximum." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "To confirm the conclusi ons, use a contour plot: (Try rotating the plot with your mouse.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "contourplot3d(f(x,y), x=-2 ..2, y=-3..3, orientation=[-90,0], axes=boxed);" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 24 "The Constrained Problem:" }}{PARA 0 "" 0 "" {TEXT -1A 97 "Find the absolute maximum and minimum values of a function insi de or on the boundary of a region." }}{PARA 5 "" 0 "" {TEXT -1 8 "Exam ple:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Extremize the function:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=(x,y)->x*y*exp(-x^2/2-y^2 /8); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*&#F/\"\"#F/*$)F.F7F/F/!\"\" *&#F/\"\")F/*$)F0F7F/F/F:F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 39 "ins ide or on the ellipse g(x,y)=1 where" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=(x,y)->x^2/4 + y^2/16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&#\"\" \"\"\"%F0*$)9$\"\"#F0F0F0*&#F0\"#;F0*$)9%F5F0F0F0F)F)F)" }}}{PARA 5 " " 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 142 "The inte rior critical points were found in the unconstrained example. There a re three methods of finding the critical points on the boundary." }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 42 "Boundary MetBhod I: Eliminate a \+ Variable " }}{PARA 0 "" 0 "" {TEXT -1 38 "Solve the constraint for one variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y0:=solve(g(x, y)=1,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G6$,$*&\"\"#\"\"\",&* $)%\"xGF(F)!\"\"\"\"%F)#F)F(F),$*&F(F)F*F0F." }}}{PARA 0 "" 0 "" {TEXT -1 212 "Notice that we named the solution y0 instead of y so tha t we can still use y as a variable. Also notice that there are 2 solu tions for the upper and lower halves of the ellipse. We must handle t hese separately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the upper half of the boundary, substitute the soluti on into the function and differentiate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f1:=MF(x,f(x,y0[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$**\"\"#\"\"\"9$F/,&*$ )F0F.F/!\"\"\"\"%F/#F/F.-%$expG6#!\"#F/F/F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "Df1:=D(f1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$Df1Gf*6C#%\"xG6\"6$%)operatorG%&arrowGF(,&*(\"\"#\"\"\",&*$)9$F.F/! \"\"\"\"%F/#F/F.-%$expG6#!\"#F/F/**F.F/F3F.F0#F4F.F7F/F4F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 90 "Find the x-coordinate at each critical po int and substitute back to find the y-coordinate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df1(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$\"\"##\"\"\"F',$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y1:=subs(x=x0[1],y0[1]); b1:=[x0[1],y1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,$*&\"\"#\"\"\"F'#F(F'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$\"\"##\"\"\"F',$*&F'F)F'F(F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y2:=subs(x=x0[2],y0[1]) ; b2:=[x0[2],y2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*&\"\"#\" \"\"F'#F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$,$*$\"\"##\" \"\"F(!\"\",$*&F(F*F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Repeat for \+ the lower half of the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f2:=MF(x,f(x,y0[2]));" }}{PARA 11 "" 1 D"" {XPPMATH 20 "6#>%#f2 Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$**\"\"#\"\"\"9$F/,&*$)F0F.F/!\" \"\"\"%F/#F/F.-%$expG6#!\"#F/F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Df2:=D(f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df2 Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*(\"\"#\"\"\",&*$)9$F.F/!\"\"\" \"%F/#F/F.-%$expG6#!\"#F/F4**F.F/F3F.F0#F4F.F7F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df2(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$\"\"##\"\"\"F',$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y3:=subs(x=x0[1],y0[2]); b3:=[x0[1] ,y3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G,$*&\"\"#\"\"\"F'#F(F'! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$*$\"\"##\"\"\"F',$*&F' F)F'F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y4:=subs(x=x 0[2],y0[2]); b4:=[x0[2],y4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y4G ,$*&\"\"#\"\"\"F'#F(F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7 $,$*$\"\"##\"\"\"F(!\"\",$*&F(F*F(F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 139 "Finally,E we tabulate the values of the function at all interior a nd boundary critical points and identify the absolute maximum and mini mum:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "p1; f&@p1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p2; f&@p2; evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$expG6#!\"\"F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p3; f&@p3; evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\" \"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$expG6#!\" \"F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p4; f&@p4; evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&\"\"#\"\"\"-%$expG6#!\"\"F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ !+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTFEXT 1 0 20 "p5; f&@p5; e valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"\"!\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$expG6#!\"\"F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b1; f&@b1; evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$\"\"##\"\"\"F%,$*&F%F'F%F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&\"\"%\"\"\"-%$expG6#!\"#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b2; f&@b2; e valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$\"\"##\"\"\"F&!\"\", $*&F&F(F&F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"\"-%$exp G6#!\"#F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G8T8a!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b3; f&@b3; evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$\"\"##\"\"\"F%,$*&F%F'F%F&!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"\"-%$expG6#!\"#F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 G0 20 "b4; f&@b4; evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$\"\"##\"\"\"F&!\"\",$*&F&F(F&F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"\"-%$expG6#!\"#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8T8a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 154 "So we see that the absolute maxima occur at the interior points (1,2) and ( -1,-2), and the absolute minima occur at the interior points (1,-2) an d (-1,2)." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 47 "Boundary Method II: Parametrize the Boundary " }}{PARA 0 "" 0 "" {TEXT -1 27 "Define th e parametrization:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r:=MF( t, [2*cos(t),4*sin(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7$f* 6#%\"tG6\"6$%)operatorG%&arrowGF),$*&\"\"#\"\"\"-%$cosG6#9$F0F0F)F)F)f *F'F)F*F),$*&\"\"%F0-%$sinGF3F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Restrict the function to the boundary:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "fr:=MF(t,simplify(f&@r(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#frGf*6#%\"tG6\"6$%)operatorG%&arrowHGF(,$**\"\")\"\" \"-%$cosG6#9$F/-%$sinGF2F/-%$expG6#!\"#F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Find the critical points on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Dfr:=D(fr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DfrGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*(\"\")\"\" \")-%$sinG6#9$\"\"#F/-%$expG6#!\"#F/!\"\"*(F.F/)-%$cosGF3F5F/F6F/F/F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bndcritpts:=solve(Df r(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+bndcritptsG6$,$*&\"\" %!\"\"%#PiG\"\"\"F),$*&F(F)F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b1:=r(bndcritpts[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$\"\"##\"\"\"F',$*&F'F)F'F(!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "b2:=r(bndcritpts[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$\"\"##\"\"\"F',$*&F'F)F'F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Since the equation is non-polynomial, solve may not give all solutions. So we plot the function for one period:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT I1 0 18 "plot(Dfr,-Pi..Pi);" }}} {PARA 0 "" 0 "" {TEXT -1 85 "From the plot and its symmetries, it is o bvious that solve missed two more solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b3:=r(3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#b3G7$,$*$\"\"##\"\"\"F(!\"\",$*&F(F*F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b4:=r(-3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$\"\"##\"\"\"F(!\"\",$*&F(F*F(F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 206 "Finally, we tabulate the values of the function at all i nterior and boundary critical points and identify the absolute maximum and minimum: \n(This was done with the first method. So we won't red o it here.)" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 "Boundary Method I II: Lagrange Multiplier Method" }}{PARA 0 "" 0 "" {TEXT -1 69 "Find \+ the gradient of the function and the gradient of the constraint:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "delf:=Grad(f)&@(x,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delfG-%'RTABLEG6%\")#R+a\"-%'VECTOJR G6#7$,$*(%\"yG\"\"\"-%$expG6#,&*&\"\"#!\"\"%\"xGF6F7*&\"\")F7F/F6F7F0, &F0F7*$)F8F6F0F0F0F7,$*&#F0\"\"%F0*(F8F0F1F0,&FAF7*$)F/F6F0F0F0F0F7&%' VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "delg:=Gr ad(g)&@(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG-%'RTABLEG6% \")GVS9-%'VECTORG6#7$,$*&\"\"#!\"\"%\"xG\"\"\"F2,$*&\"\")F0%\"yGF2F2&% 'VectorG6#%$rowG" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Set up the Lagrange \+ multiplier equations and solve for the critical points on the boundary :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqs:=equate(delf,simpli fy(lambda*delg));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,$*(%\" yG\"\"\"-%$expG6#,&*&\"\"#!\"\"%\"xGF0F1*&\"\")F1F)F0F1F*,&F*F1*$)F2F0 F*F*F*F1,$*(F0F1%'lambdaGF*F2F*F*/,$*&#F*\"\"%F**(F2F*F+F*,&F?F1*$)F)F 0F*F*F*F*F1,$*(F4F1F:F*F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol:=solve(\{op(eqs),g(x,y)=1\},\{x,y,lambda\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG6$<%/%'lambdaG,$*&\"\"%\"\"\"-%$expG6#!\"#F ,!\"\"/%\"yG,$*&\"\"#F,K-%'RootOfG6$,&*$)%#_ZGF6F,F,F6F1/%&labelG%$_L5G F,F,/%\"xGF7<%F2/FB,$F7F1/F(,$*&F+F,F-F,F," }}}{PARA 0 "" 0 "" {TEXT -1 70 "Since the solutions involve a RootOf, we resolve them using all values:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=allvalues(s ol[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G6$<%/%'lambdaG,$*& \"\"%\"\"\"-%$expG6#!\"#F,!\"\"/%\"xG*$\"\"##F,F5/%\"yG,$*&F5F,F5F6F,< %F'/F3,$F4F1/F8,$*&F5F,F5F6F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1:=subs(sol1[1],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b 1G7$*$\"\"##\"\"\"F',$*&F'F)F'F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b4:=subs(sol1[2],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$\"\"##\"\"\"F(!\"\",$*&F(F*F(F)F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol2:=allvalues(sol[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G6$<%/%\"yG,$*&\"\"#\"\"\"F+#F, F+F,/%\"xG,$*$F+F-!\"\"/%'lambdaG,$*&\"\"%F,-%$expG6#!\"#F,F,<%/F/F1/F (,$*&F+F,F+F-F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b3:=suL bs(sol2[1],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$,$*$\"\" ##\"\"\"F(!\"\",$*&F(F*F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2:=subs(sol2[2],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b 2G7$*$\"\"##\"\"\"F',$*&F'F)F'F(!\"\"" }}}{EXCHG }{PARA 0 "" 0 "" {TEXT -1 208 "Finally, we tabulate the values of the function at all i nterior and boundary critical points and identify the absolute maximum and minimum: \n (This was done with the first method. So we won't r edo it here.)" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1 995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "equate" 2 "equate" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "so lve" 2 "solve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fsolve" 2 "fsolve" " " }{TEXT -1 2 ", " }{HYPERLNK M17 "allvalues" 2 "allvalues" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "LeadingPrincipalMinorDeterminants" 2 "LeadingPrincipalMi norDeterminants" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "op" 2 "op" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "MakeFunction" 2 "MakeFunction" "" }{TEXT -1 2 ". " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 14368276 12934324 14277132 14534024 15096356 2473708 15479156 15400392 14404328 }{RTABLE M7R0 I5RTABLE_SAVE/14368276X*%)anythingG6"6"[gl!$%!!!"#"#f*6$%"xG%"yGF&6$%)operatorG %&arrowGF&,$*(9%"""-%$expG6#,&*$9$""##!""F8*$F0F8#F:"")F1,&F:F1F6F1F1F:F&F&F&f* F(F&F+F&,$*(F7F1F2F1,&!"%F1F;F1F1#F:""%F&F&F&F& } {RTABLE M7R0 I5RTABLE_SAVE/12934324X,%)anythingG6"6"[gl!"%!!!#%"#"#**%"xG"""%"yGF)-%$expG6#, &*$F(""##!""F0*$F*F0#F2"")F),&!"$F)F/F)F),$*&F+F),*""%F)F/!"%F3F2*&F(F0F*F0F)F) #F)F;F8,$**F(F)F*F)F+F),&!#7F)F3F)F)#F)"#;F& } {RTABLE M7R0N I5RTABLE_SAVE/14277132X,%)anythingG6"6"[gl!"%!!!#%"#"#""!"""F(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/14534024X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""!"%""!F-,$F (F+F& } {RTABLE M7R0 I5RTABLE_SAVE/15096356X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""""%""!F-F(F & } {RTABLE M7R0 I4RTABLE_SAVE/2473708X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""""%""!F-F(F& } {RTABLE M7R0 I5RTABLE_SAVE/15479156X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""!"%""!F-,$F (F+F& } {RTABLE M7R0 I5RTABLE_SAVE/15400392X*%)anythingG6"6"[gl!$%!!!"#"#,$*(%"yG"""-%$expG6#,&*$%"x G""##!""F1*$F)F1#F3"")F*,&F3F*F/F*F*F3,$*(F0F*F+F*,&!"%F*F4F*F*#F3""%F& } {RTABLE M7R0 I5RTABLE_SAVE/14404328X*%)anythingG6"6"[gl!$%!!!"#"#,$%"xG#"""""#,$%"yG#F*"")F& } * F(F&F+F&,$*(F7F1F2F1,&!"%F1F;F1F1#F:""%F&F&F&F& } {RTABLE M7R0 I5RTABLE_SAVE/12934324X,%)anythingG6"6"[gl!"%!!!#%"#"#**%"xG"""%"yGF)-%$expG6#, &*$F(""##!""F0*$F*F0#F2"")F),&!"$F)F/F)F),$*&F+F),*""%F)F/!"%F3F2*&F(F0F*F0F)F) #F)F;F8,$**F(F)F*F)F+F),&!#7F)F3F)F)#F)"#;F& } {RTABLE M7R0Nmcconnel"md"me"measur""""memb"memor ""mension"mes"""""method"mf7"""""""""""""mfg"mg ""min"minant"mini"minima"minimum ""minor """miss"mmand"mng ""modifi"modul"mont"more ""most ""mous"mu ""mui"muin"muint"""muintg"mul ""mulitipl"multidimensional" multimaxmi" multimaxmin """""multipl """"" multiplei" multiplein" multipleint'$"""""""""multipli" multivaria" multivariabl ""mum"must#""""""""na ""nal"nam ""nameg%"""""""""""""""""""""""""named"nant"nate ""nce ""nces"ncipalminordeterminant ""nction"""nd""""ndrical"nds"sk"sn""""sng"snl"""snlg"sol " solg"solu"solut"soluti"solution"solv"som ""some ""spac"""" spacecurv"spe"spec ""specif ""specifi#"""""""" specificat/""""""""""" specificati ""spect"speed ""sph$"""sphe"spher"spheri" spherical " spiral"splay"spot""""ing ""ingf"initial"input,""""""""""""""""""""""""""""""""""""ins"insi"instead"""""""int#""""""""inte ""integr""""integra"integral*"""" "" integrand"integrat"""""intel$""""""""""""""""""""""""""""""""""""interior"interm" intermediat"""""internal" interpret"intg"""""""into "" introduct" intvector"equat"equation"equival""""er""""erat"eratorg ""ere"ergenc"erridden"ersion"es;""""""""""""""ese" essential"ession"""etagf"eter"etermin" eterminant ""ethan"etors"etween"ev"eva"evalf """"""evalfu" evalfunct'"""" """""cosg7/""" """"""""""cosgf/"""""""""""cosgfbf"couL"""""""""""""""""""""""""""""""""""counterclockwis ""cour""""couri"" "" """"" """" """"""""="(""""""" """""" " cpn"cr"cript"critical ""critpt"critptsg"cro"cros"""ygfcritptolvecritptsgdeterminaximumnotemayfailhfhfgrtablegcvmatrixgfffsimplifyexpgfdfkfatrincipalinterpretresultlpmkrfsincnegatcspositcjmatrithirdpzfourthpointzafifthconfirmconclusionscontourplottryrotatyourmouscontourplotorientataxesboxedabsolutvaluinsideboundarregionexampleextremizoperatorginsideellipsggfinteriorwerefoundremethodeliminatvariablconstraintnoticwenamedinsteadthastillalsosoluupperlowerhalvhandlheseseparatehalfsubstitutdifferentiatgfdfbackrepeatfinaltabulatinteriorndidentifminimumevalfedtvalfmaximaoccurminimaiiparametrizparametrizatcossinrgtgcosgsingfrestrictfrsimpliffrgfdfrdfrgfsingcosgfbndcritptbndcritptsgpignonpolynomialgiveperiodpisymmetrbvioumissmorenteriordonewonredherevectorfafdelggraddelgggvssimplifylambdalambdagfsolglambdagrootofgzgflabelgfbinvolvresolvthemolsubsedocopyrightarthurbelmontphilipyasskindepartmmathVematictexauniversitlveleadingprincipalminordeterminantsaveanythinggglxgfffsimplifyexpgfdfkfatrincipalinterpretresultlpmkrfsincnegatcspositcjmatrithirdpzfourthpointzafifthconfirmconclusionscontourplottryrotatyourmouscontourplotorientataxesboxedabsolutvaluinsideboundarregionexampleextremizoperatorginsideellipsggfinteriorwerefoundremethodeliminatvariablconstraintnoticwenamedinsteadthastillalsosoluupperlowerhalvhandlheseseparatehalfsubstitutdifferentiatgfdfbackrepeatfinaltabulatinteriorndidentifminimumevalfedtvalfmaximaoccurminimaiiparametrizparametrizatcossinrgtgcosgsingfrestrictfrsimpliffrgfdfrdfrgfsingcosgfbndcritptbndcritptsgpignonpolynomialgiveperiodpisymmetrbvioumissmorenteriordonewonredherevectorfafdelggraddelgggvssimplifylambdalambdagfsolglambdagrootofgzgflabelgfbinvolvresolvthemolsubsedocopyrightarthurbelmontphilipyasskindepartmmathV"${VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLXE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "TimYes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 259 20 "VecCalc[Gradient] - " }{TEXT -1 55 "Calculates the Gr adient of a Function in Arrow Notation" }}{PARA 0 "" 0 "" {TEXT 26 6 " Alias:" }{TEXT -1 48 " - The alias can be used after execution of the \+ " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT 276 20 " Grad = Gradient" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }}{PARA 0 "" 0 "" {TEXT 256 92 " Gr adient(f, vars, outtype) Grad(f, vars, outtype) VecCalc[Gradient]( f, vars, outtype)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 13 " f - " }{TEXT -1 74 " a Zscalar function in the form of an arrow-defined function of n variab les." }}{PARA 0 "" 0 "" {TEXT 258 13 " vars - " }{TEXT -1 1 "(" } {TEXT 262 8 "optional" }{TEXT -1 89 ") sequence, list or Vector of nam es to be used as independent variables for the function." }}{PARA 0 " " 0 "" {TEXT 260 13 " outtype - " }{TEXT 278 24 "(optional) output t ype: " }{TEXT 279 51 "'list', 'Vector', 'Vector[row]' or 'Vector[colum n]'" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 86 "The gradient of a function is th e vector of first partial derivatives of the function." }}{PARA 15 "" 0 "" {TEXT 261 8 "Gradient" }{TEXT -1 124 " acts on an arrow-defined f unction and returns a list or Vector of the first partial derivatives \+ as arrow-defined functions." }}{PARA 15 "" 0 "" {TEXT -1 207 "The spec ification of variables is optional unless Maple is unable to determine the variables for the function. This can happen if the function is co nstant, is a built-in functi[on or is an undefined function." }}{PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 263 7 "outtype" }{TEXT -1 84 " \+ parameter is specified, then the output is converted to have the type \+ specified by " }{TEXT 264 7 "outtype" }{TEXT -1 64 ". Otherwise the ou tput type is specified by the global variable " }{HYPERLNK 17 "OutputV ectorType" 2 "OutputVectorType" "" }{TEXT -1 28 " which by default is \+ set to " }{TEXT 265 13 "'Vector[row]'" }{TEXT 266 0 "" }{TEXT 267 1 ". " }}{PARA 15 "" 0 "" {TEXT 268 8 "Gradient" }{TEXT -1 26 " differs fro m the command " }{HYPERLNK 17 "linalg[grad]" 2 "linalg[grad]" "" } {TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 122 " package which acts on an expression and returns a vector of expr essions. It requires the specification of the variables." }}{PARA 15 "" 0 "" {TEXT 269 8 "Gradient" }{TEXT -1 26 " differs from the command " }{HYPERLNK 17 "VectorCalculus[Gradient]" 2 "VectorCalculus[Gradient ]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorCalc\ulus" 2 "VectorC alculus" "" }{TEXT -1 175 " package which acts on an expression and re turns a VectorField of expressions. The VectorCalculus package requir es the specification of a coordinate system and its variables." }} {PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 272 8 "Gradient" } {TEXT -1 16 " is part of the " }{TEXT 271 7 "VecCalc" }{TEXT -1 41 " p ackage, and so can be used in the form " }{TEXT 270 8 "Gradient" } {TEXT -1 35 " only after performing the command " }{TEXT 273 13 "with( VecCalc)" }{TEXT -1 4 " or " }{TEXT 274 23 "with(VecCalc, Gradient)" } {TEXT -1 56 ". The function can always be accessed in the long form \+ " }{TEXT 275 17 "VecCalc[Gradient]" }{TEXT -1 13 ". The alias " } {TEXT 277 4 "Grad" }{TEXT -1 47 " can be used only after performing th e command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {EXCHG {PARA 0 "> " 0 "" {]MPLTEXT 1 0 24 "f:=(x,y,z)->x^2*y^3*z^4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%)operat orG%&arrowGF**()9$\"\"#\"\"\")9%\"\"$F2)9&\"\"%F2F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "delf:=Gradient(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delfG-%'RTABLEG6%\")+s\"H\"-%'VECTORG6#7%f*6%% \"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF2,$**\"\"#\"\"\"9$F9)9%\"\"$F9) 9&\"\"%F9F9F2F2F2f*F.F2F3F2,$**F=F9)F:F8F9)FF9F9F2F2F2f*F.F2F3F 2,$**F@F9FDF9F;F9)F?F=F9F9F2F2F2&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalFunction(delf,x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")gt\"H\"-%'VECTORG6#7%,$**\"\"#\"\"\"% \"xGF.)%\"yG\"\"$F.)%\"zG\"\"%F.F.,$**F2F.)F/F-F.)F1F-F.F3F.F.,$**F5F. F8F.F0F.)F4F2F.F.&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=MakeFunction([x,y],2*x^2*y+exp(y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),& *(\"\"#\"\"\")9$F/F09%F0F0-%$expG6#F3F0F)F)F)" }}}{EXCHG {PARA ^0 "> " 0 "" {MPLTEXT 1 0 36 "delg:=Grad(g,[a,b], Vector[column]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG-%'RTABLEG6%\")oPP8-%'MATRIXG6#7$7#f *6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF2,$*(\"\"%\"\"\"9%F99$F9F9F2F2F2 7#f*F/F2F3F2,&*&\"\"#F9)F;F@F9F9-%$expG6#F:F9F2F2F2&%'VectorG6#%'colum nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "EF(delg,);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")))QP8-%'MATRIXG6#7$7#,$ *(\"\"%\"\"\"%\"qGF/%\"pGF/F/7#,&*&\"\"#F/)F1F5F/F/-%$expG6#F0F/&%'Vec torG6#%'columnG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright \+ 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department o f Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "Se e Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "linalg[grad]" 2 "linalg[grad]" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[Gradient]" 2 "LinearAlgeb ra[Gradient]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLN_K 17 "MultiMaxMin" 2 "MultiMaxMin" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Divergence" 2 "Divergence" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " ScalarPotential" 2 "ScalarPotential" "" }{TEXT -1 2 ". " }}}}{MARK "0 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12917200 12917360 13373768 13373888 }{RTABLE M7R0 I5RTABLE_SAVE/12917200X*%)anythingG6"6"[gl!$%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,$*(9$"""9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(F2F8F4F8F6F7F5F,F,F,f*F (F,F-F,,$*(F2F8F4F5F6F5F7F,F,F,F, } {RTABLE M7R0 I5RTABLE_SAVE/12917360X*%)anythingG6"6"[gl!$%!!!"$"$,$*(%"xG"""%"yG""$%"zG""%"" #,$*(F)F/F+F/F-F.F,,$*(F)F/F+F,F-F,F.6" } {RTABLE M7R0 I5RTABLE_SAVE/13373768X*%)anythingG6"6"[gl!#%!!!"#"#f*6$%"aG%"bG6"6$%)operatorG %&arrowGF+,$*&9%"""9$F2""%F+F+F+f*F(F+F,F+,&*$F3""#F8-%$expG6#F1F2F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/13373888X*%)anythingG6"6"[gl!#%!!!"#"`#,$*&%"qG"""%"pGF*""%,&*$F+" "#F/-%$expG6#F)F*6" } " }{HYPERLNK 17 "Divergence" 2 "Divergence" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Hessian" 2 "Hessian" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " ScalarPotential" 2 "ScalarPotential" "" }{TEXT -1 2 ". " }}}}{MARK "0 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12917200 12917360 13373768 13373888 }{RTABLE M7R0 I5RTABLE_SAVE/12917200X*%)anythingG6"6"[gl!$%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,$*(9$"""9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(F2F8F4F8F6F7F5F,F,F,f*F (F,F-F,,$*(F2F8F4F5F6F5F7F,F,F,F, } {RTABLE M7R0 I5RTABLE_SAVE/12917360X*%)anythingG6"6"[gl!$%!!!"$"$,$*(%"xG"""%"yG""$%"zG""%"" #,$*(F)F/F+F/F-F.F,,$*(F)F/F+F,F-F,F.6" } {RTABLE M7R0 I5RTABLE_SAVE/13373768X*%)anythingG6"6"[gl!#%!!!"#"#f*6$%"aG%"bG6"6$%)operatorG %&arrowGF+,$*&9%"""9$F2""%F+F+F+f*F(F+F,F+,&*$F3""#F8-%$expG6#F1F2F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/13373888X*%)anythingG6"6"[gl!#%!!!"#"`fgfhf"fgfqf"fh"fhf"""fi"field/1"""""""""""fif"fifth"fin"final"find ""fined"finishi"first;""""""""""""""fix"fixe ""fixed3""""""""""""fkf ""float"""flux"fn"fnc """fnf"fnfcfgf"fng"fo ""follow ""fops"forget"formF""""""""""""""""""""""""""""""""formula ""found"fourth"""fpf"fr ""frenet"""frf"frgf"fro"fsolv"fter"ftf"fu"""fun"""func"""funct""" """"""""""""""" """" " """"""""functi ""functio"""""functioncD""""""""""""""""""""""""upper"upward ""urfac"urfaceintscalarinert"urfaceintvector ""url" urvecurvatur"us"""use'"""""""""usedS""""""""""""""""""""""""""""""""user"uses"""""usin"usingC!""""""""""""""""usual"ut"utes ""utput ""uts"va ""veccal """""""veccalcf" """ " " " "" " " """ """"""""" "" " """" "" " " " " vecpot"vect"vecto ""vectorw""""""""""" """""" "" """""""""" vectorangl"vectorc"""vectorca" vectorcal" vectorcalcu "" vectorcalcul "" vectorcalculu7J""""""""""""" vectorfield """"""vectorgkV"""""""""""""""""""""""""" vectorinert" vectornorm" vectorpot ""algebralinearalgebradiffopmultimaxmindivergenccurlhessianscalarpotentialrtablsaveanythinggglqgitemfunctveccalcgradicalculatgradientarrownotataliacanusedafterexecutvcaliacommandgradcallsequencvarsouttypparameterscalarformdefinvariablesoptionallistvectornamesindependvariablyperowcolumdescriptthfirstpartialderivatactsunctreturnfunctionspecificatunlesunabldeterminhappenconstantbuiltundefinparametspecificonverthavetypeotherwisoutputglobaloutputvectortypoutputvectortypdefaultsetdifferfrolinalgpackagexpressexpressionrequirspecificatvectorcalculuvectorcalculureturnvectorfieldcoordinatsystempartackagonlyperformwithalwayaccesslongexamplfgfxgygzgoperatorgarrowgfdelfdelfgrtablegvectorgoperatorgfdfrowgevalfunctgtxgfmakefunctexpggfexpgdelgcolumndelggoppmatrixgagbgngefqpqgfpgfvectorgcolumngcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitsealsolineare""{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1g 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Timhes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 257 19 "VecCalc[Hessian] - " }{TEXT -1 55 "Calculates the Hes sian of a Function in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 6 "A lias:" }{TEXT -1 48 " - The alias can be used after execution of the \+ " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT 258 19 " Hess = Hessian" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }}{PARA 0 "" 0 "" {TEXT 26 0 "" } {TEXT 256 92 " Hessian(f, vars, outtype) Hess(f, vars, outtype) \+ VecCalc[Hessian](f, vars, outtype)" }}{PARA 0 "" 0 "" {TEXT 26 11 "P arameters:" }}{PARA 0 "" 0 "" {TEXT 259 13 " f - " }{TEXT -1 74 "a scalar function in the form of an arrow-defined function of n va riables." }}{PARA 0 "" 0 "" {TEXT 260 13 " vars i- " }{TEXT -1 1 " (" }{TEXT 262 8 "optional" }{TEXT -1 89 ") sequence, list or Vector of names to be used as independent variables for the function." }}{PARA 0 "" 0 "" {TEXT 261 13 " outtype - " }{TEXT 263 24 "(optional) outpu t type: " }{TEXT 264 22 "'listlist' or 'Matrix'" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 86 "The Hessian of a function is the matrix of second partial derivatives of the function." }}{PARA 15 "" 0 "" {TEXT 265 7 "Hessian " }{TEXT -1 134 " acts on an arrow-defined function and returns a list of lists or Matrix of the second partial derivatives as arrow defined functions." }}{PARA 15 "" 0 "" {TEXT -1 207 "The specification of var iables is optional unless Maple is unable to determine the variables f or the function. This can happen if the function is constant, is a bui lt-in function or is an undefined function." }}{PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 266 7 "outtype" }{TEXT -1 84 " parameter is spec ified, thejn the output is converted to have the type specified by " } {TEXT 267 7 "outtype" }{TEXT -1 64 ". Otherwise the output type is spe cified by the global variable " }{HYPERLNK 17 "OutputMatrixType" 2 "Ou tputMatrixType" "" }{TEXT -1 28 " which by default is set to " }{TEXT 268 8 "'Matrix'" }{TEXT 269 0 "" }{TEXT 270 1 "." }}{PARA 15 "" 0 "" {TEXT 272 7 "Hessian" }{TEXT -1 26 " differs from the command " } {HYPERLNK 17 "linalg[hessian]" 2 "linalg[hessian]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 122 " package wh ich acts on an expression and returns a matrix of expressions. It req uires the specification of the variables." }}{PARA 15 "" 0 "" {TEXT 271 7 "Hessian" }{TEXT -1 26 " differs from the command " }{HYPERLNK 17 "VectorCalculus[Hessian]" 2 "VectorCalculus[Hessian]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorCalculus" 2 "VectorCalculus" "" } {TEXT -1 146 " package which acts on an expression and returns a Matri x of expressions. The VectorCalculus package reqkuires the specificati on of the variables." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " } {TEXT 273 7 "Hessian" }{TEXT -1 16 " is part of the " }{TEXT 274 7 "Ve cCalc" }{TEXT -1 41 " package, and so can be used in the form " } {TEXT 275 7 "Hessian" }{TEXT -1 35 " only after performing the command " }{TEXT 276 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 277 22 "wit h(VecCalc, Hessian)" }{TEXT -1 56 ". The function can always be acces sed in the long form " }{TEXT 278 16 "VecCalc[Hessian]" }{TEXT -1 13 " . The alias " }{TEXT 279 4 "Hess" }{TEXT -1 47 " can be used only aft er performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" } {TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCal c): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=(x,y,z)- >x^2*y^3*z^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG% \"zG6\"6$%)operatorG%&arrowGF**()9$\"\"#\"\"\")9%\"\"$F2)9&\"\"%F2F*F* F*" }}}{EXCHG {PlARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Hf:=Hessian(f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HfG-%'RTABLEG6%\")%32M\"-%'MATRIXG6 #7%7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3,$*(\"\"#\"\"\")9%\" \"$F:)9&\"\"%F:F:F3F3F3f*F/F3F4F3,$**\"\"'F:9$F:)FF:F:F3F3F3f*F /F3F4F3,$**\"\")F:FEF:F;F:)F?F=F:F:F3F3F37%FAf*F/F3F4F3,$**FDF:)FEF9F: FF:F:F3F3F3f*F/F3F4F3,$**\"#7F:FPF:FFF:FKF:F:F3F3F37%FGFQf*F/F3F4 F3,$**FTF:FPF:F;F:)F?F9F:F:F3F3F3%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "evalFunction(Hf,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")[%4M\"-%'MATRIXG6#7%7%,$*(\"\"#\"\"\")% \"bG\"\"$F/)%\"cG\"\"%F/F/,$**\"\"'F/%\"aGF/)F1F.F/F3F/F/,$**\"\")F/F9 F/F0F/)F4F2F/F/7%F6,$**F8F/)F9F.F/F1F/F3F/F/,$**\"#7F/FBF/F:F/F>F/F/7% F;FC,$**FEF/FBF/F0F/)F4F.F/F/%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=MakeFunction([x,y],2*x^2*y+exp(y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),& *(\"\"#\"\"\")9$F/F09%F0F0-%$expG6#F3F0F)F)F)" }}}{EXmCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Hg:=Hess(g,[a,b],listlist);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HgG7$7$f*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF+,$*& \"\"%\"\"\"9%F2F2F+F+F+f*F(F+F,F+,$*&F1F29$F2F2F+F+F+7$F4f*F(F+F,F+-%$ expG6#F3F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Hg(p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,$*&\"\"%\"\"\"%\"qGF(F(,$*&F'F( %\"pGF(F(7$F*-%$expG6#F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- C opyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Dep artment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecC alc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[hessian]" 2 "linalg[hes sian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LinearAlgebra[Hessian]" 2 "L inearAlgebra[Hessian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 " Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MultiMaxMin" 2 "MultiMaxMi n" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" } {TEXT -1 2 ", " }{HnYPERLNK 17 "Laplacian" 2 "Laplacian" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LeadingPrincipalMinorDeterminants" 2 "LeadingPri ncipalMinorDeterminants" "" }{TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 13407084 13409448 }{RTABLE M7R0 I5RTABLE_SAVE/13407084X,%)anythingG6"6"[gl!"%!!!#*"$"$f*6%%"xG%"yG%"zG6"6$%)ope ratorG%&arrowGF,,$*&9%""$9&""%""#F,F,F,f*F(F,F-F,,$*(9$"""F2F6F4F5""'F,F,F,f*F( F,F-F,,$*(F:F;F2F3F4F3"")F,F,F,F7f*F(F,F-F,,$*(F:F6F2F;F4F5F " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "F:=MakeFunction([x,y,z],);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'RTABLEG6%\")g`\"H\"-%'VECTOR G6#7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF2,(*$)9$\"\"#\"\"\"F; *$)9%\"\"$F;F;*(F9F;F>F;)9&\"\"%F;zF;F2F2F2f*F.F2F3F2*&)F>F:F;FBF;F2F2F 2f*F.F2F3F2*&F8F;)FBF?F;F2F2F2&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Divergence(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF(,**&\"\"#\"\"\"9$F/ F/*&9%F/)9&\"\"%F/F/*(F.F/F2F/F4F/F/*(\"\"$F/)F0F.F/)F4F.F/F/F(F(F(" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthu r Belmonte and Philip B. Yasskin\n Department of Mathematics, Tex as A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "linalg[diverge]" 2 "linalg[diverge]" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "VectorCalculus[Divergence]" 2 "VectorCalculus[Diverge nce]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "V ectorPotential" 2 "VectorPotential" "" }{TEXT -1 2 ". " }}}}{MARK "0 0{ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 12915360 }{RTABLE M7R0 I5RTABLE_SAVE/12915360X*%)anythingG6"6"[gl!$%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,(*$9$""#"""*$9%""$F4*(F2F4F6F49&""%F4F,F,F,f*F(F,F-F,*&F6F3F9F4F ,F,F,f*F(F,F-F,*&F2F3F9F7F,F,F,F, } F/F(F(F(" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthu r Belmonte and Philip B. Yasskin\n Department of Mathematics, Tex as A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "linalg[diverge]" 2 "linalg[diverge]" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "VectorCalculus[Divergence]" 2 "VectorCalculus[Diverge nce]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "V ectorPotential" 2 "VectorPotential" "" }{TEXT -1 2 ". " }}}}{MARK "0 0{involv ""ion# """"""""ional"ions ""ip"ipal"iption"ishing"isplay ""ists"item%""""""""""""""""""""""""""""""""""""ith"ithout"iv ""ix"ixg ""ja"jac""""jacg"jaco"jacobi"jacobia"jacobian""" " jacobiand" jacobiandet"jacobiandeterminan"jacobiandeterminant"""""" jacobianm" jacobianmat" jacobianmatr"jacobianmatrix"""""jame"jared"jdet""""jdetg"jeffre ""jerk"js ""jsg"jsgf"jt ""jtg"jtgf"just"kage""""ke"ken"keu"kf"kg"krf"krista"kw"la ""labelg"lagrang"lambda"lambdag"lambdagf"lap""""U{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 16 "VecCalc[Curl] - " }{TEXT -1 74 "Calculates the Curl o f a Three-Dimensional Vector Field in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 61 " Curl(F, vars, outtype) VecCalc[Curl](F, vars, out type)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 256 13 " F - " }{TEXT -1 105 "a 3-dimens ional vector field in the form of a list or Vector of 3 arrow-defined \+ functions of 3 variables " }}{PARA 0 "" 0 "" {TEXT 259 13 " vars \+ - " }{TEXT -1 1 "(" }{TEXT 260 8 "optional" }{TEXT -1 91 ") sequence, \+ list or Vector of 3 names to be used as independent variables for the \+ function." }}{PARA 0 "" 0 "" {TEXT 261 13 " outtype - " }{TEXT 262 24 "(optional) output type: " }{TEXT 263 51 "'list', 'Vector', 'Vector [row]' or 'Vector[column]'" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "De scription:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 267 4 "Curl" } {TEXT -1 156 " acts on a list or Vector of 3 arrow-defined functions \+ of 3 variables and returns its curl as a list or Vector of 3 arrow-de fined function of 3 variables." }}{PARA 15 "" 0 "" {TEXT -1 212 "The s pecification of variables is optional unless Maple is unable to determ ine the variables for the vector field. This can happen if the functio ns are constant, are built-in functions or are undefined functions." } }{PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 264 7 "outtype" }{TEXT -1 84 " parameter is specified, then the output is converted to have t he type specified by " }{TEXT 265 7 "outtype" }{TEXT -1 65 ". Otherwis e the output type matches the type of the vector field " }{TEXT 266 1 "F" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 268 4 "Curl" }{TEXT -1 26 " differs from the command " }{HYPERLNK 17 "linalg[curl]" 2 "linalg [curl]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" " " }{TEXT -1 145 " package which acts on a list or vector of 3 expressi ons and returns a vector of 3 expressions. It requires the specificat ion of the variables. " }}{PARA 15 "" 0 "" {TEXT 271 4 "Curl" }{TEXT -1 26 " differs from the command " }{HYPERLNK 17 "VectorCalculus[Curl] " 2 "VectorCalculus[Curl]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "Ve ctorCalculus" 2 "VectorCalculus" "" }{TEXT -1 194 " package which acts on a VectorField of 3 expressions and returns a VectorField of 3 expr essions. The VectorCalculus package requires the specification of a c oordinate system and its variables." }}{PARA 15 "" 0 "" {TEXT -1 13 "T he function " }{TEXT 269 4 "Curl" }{TEXT -1 16 " is part of the " } {TEXT 272 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in th e form " }{TEXT 270 4 "Curl" }{TEXT -1 35 " only after performing the \+ command " }{TEXT 273 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 274 19 "with(VecCalc, Curl)" }{TEXT -1 56 ". The function can always be a ccessed in the long form " }{TEXT 275 13 "VecCalc[Curl]" }{TEXT -1 1 " ." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "F:=MakeFunction([x,y,z],[x^2 +y^3+x*y*z^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F 4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+f*F'F+F,F+*&)F7F3F4F;F4F+F+ F+f*F'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "CF:=Curl(F); CF(a,b,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# CFG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,$*$)9%\"\"#\"\"\"! \"\"F+F+F+f*F'F+F,F+,&**\"\"%F49$F4F2F4)9&\"\"$F4F4*(F3F4F:F4F;F4F5F+F +F+f*F'F+F,F+,&*&F=F4F1F4F5*&F:F4)F " 0 "" {MPLTEXT 1 0 33 "CF:=Curl(F,Vector); EF(CF,a,b,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CFG-%'RTABLEG6%\")SS\"H\"-%'MATRI XG6#7%7#f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3,$*$)9%\"\"#\"\" \"!\"\"F3F3F37#f*F/F3F4F3,&**\"\"%F<9$F " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(x,y,z)->x^2+y^3+x*y*z^4; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%)op eratorG%&arrowGF*,(*$)9$\"\"#\"\"\"F3*$)9%\"\"$F3F3*(F1F3F6F3)9&\"\"%F 3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Lf:=Lap(f); Lf (a,b,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LfGf*6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF*,(\"\"#\"\"\"*&\"\"'F09%F0F0**\"#7F09$F0F3F0) 9&F/F0F0F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"#\"\"\"*&\"\"' F%%\"bGF%F%**\"#7F%%\"aGF%F(F%)%\"cGF$F%F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "F:=MakeFunction([x,y,z],);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'RTABLEG6%\"))ovL \"-%'MATRIXG6#7%7#f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3,(*$)9$ \"\"#\"\"\"F<*$)9%\"\"$F " 0 "" {MPLTEXT 1 0 37 "LapF:=Laplacian(F); EF(La pF,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LapFG-%'RTABLEG6%\" )))eP8-%'MATRIXG6#7%7#f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3,( \"\"#\"\"\"*&\"\"'F99%F9F9**\"#7F99$F9F " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T:=MakeFunction(,); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'RTABLEG6%\")op,6-%'MATRIX G6#7%7#f*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF2,&*$)9$\"\"#\"\"\"F;*$) 9%F:F;F;F2F2F27#f*F/F2F3F2,&F9F;F>F;F2F2F27#f*F/F2F3F2*&F>F;F9F;F2F2F2 &%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "JT :=JacobianMatrix(T); evalFunction(JT,a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#JTG-%'RTABLEG6%\")#\\HM\"-%'MATRIXG6#7%7$f*6$%\"uG% \"vG6\"6$%)operatorG%&arrowGF2,$*&\"\"#\"\"\"9$F9F9F2F2F2f*F/F2F3F2,$* &F8F99%F9F9F2F2F27$F9F97$f*F/F2F3F2F>F2F2F2f*F/F2F3F2F:F2F2F2%'MatrixG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")3PV8-%'MATRIXG6#7%7 $,$*&\"\"#\"\"\"%\"aGF/F/,$*&F.F/%\"bGF/F/7$F/F/7$F3F0%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "S:=MF([rho,theta,phi], [rho* sin(phi)*cos(theta),\nrho*sin(phi)*sin(theta), rho*cos(phi)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7%f*6%%$rhoG%&thetaG%$phiG6\"6$% )operatorG%&arrowGF+*(9$\"\"\"-%$sinG6#9&F1-%$cosG6#9%F1F+F+F+f*F'F+F, F+*(F0F1F2F1-F3F8F1F+F+F+f*F'F+F,F+*&F0F1-F7F4F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "JS:=Jac(S,[r,t,p]); JS(3,Pi/2,Pi/2) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#JSG7%7%f*6%%\"rG%\"tG%\"pG6\"6 $%)operatorG%&arrowGF,*&-%$sinG6#9&\"\"\"-%$cosG6#9%F5F,F,F,f*F(F,F-F, ,$*(9$F5F1F5-F2F8F5!\"\"F,F,F,f*F(F,F-F,*(F=F5-F7F3F5F6F5F,F,F,7%f*F(F ,F-F,*&F1F5F>F5F,F,F,f*F(F,F-F,*(F=F5F1F5F6F5F,F,F,f*F(F,F-F,*(F=F5FBF 5F>F5F,F,F,7%f*F(F,F-F,FBF,F,F,\"\"!f*F(F,F-F,,$*&F=F5F1F5F?F,F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%7%\"\"!!\"$F%7%\"\"\"F%F%7%F%F%F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "JacobianDeterminant(S);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6%%$rhoG%&thetaG%$phiG6\"6$%)operat orG%&arrowGF(,$*&-%$sinG6#9&\"\"\")9$\"\"#F2!\"\"F(F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " linalg[jacobian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VectorCalculus[Jacobian]" 2 "VectorCalculus[Jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" } {TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 11016968 13429492 13433708 }{RTABLE M7R0 I5RTABLE_SAVE/11016968X*%)anythingG6"6"[gl!#%!!!"$"$f*6$%"uG%"vG6"6$%)operatorG %&arrowGF+,&*$9$""#"""*$9%F2F3F+F+F+f*F(F+F,F+,&F1F3F5F3F+F+F+f*F(F+F,F+*&F5F3F 1F3F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/13429492X,%)anythingG6"6"[gl!"%!!!#'"$"#f*6$%"uG%"vG6"6$%)operato rG%&arrowGF+,$9$""#F+F+F+"""f*F(F+F,F+9%F+F+F+f*F(F+F,F+,$F4F1F+F+F+F2f*F(F+F,F +F0F+F+F+F+ } {RTABLE M7R0 I5RTABLE_SAVE/13433708X,%)anythingG6"6"[gl!"%!!!#'"$"#,$%"aG""#"""%"bG,$F+F)F*F (6" } bian]" 2 "VectorCalculus[Jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" } {TEXT -1 2 ". " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGEsurfacenormallength "" surfacetang"surfacetangent""""sw"symbol ""symbolic"symmetr"syn"system' """""""""ta"""tabl """"tableg"tabulat"tang""" tangential"te ""ted"tempt"tential"termin" terminant"ters ""tes ""teslo"test"tex"texa#"""""""""""""""""""""""""""""""""""text"tg3D""""""""""""tgf """""thO"""""""""""""""""""tha"their"""""them ""thesS""""""""""""""""""""thet"thetaV"""&""" "jsgpgfbfjacobiandeterminantoperatorgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitdiffopmultipleintcoordconversrtablsaveanythinggglagbgattransformatparametrizedsurfacspacaliacanusedafterexecutvcaliacommandjaccallsequencjacobianmatrixvarsouttypjacobianmatrixparameterformlistvectorarrowdefinfunctionvariabloptionalnameindependariabltypelistlistdescriptrgkgngwhosijthntrypartialderivatcomponwithrespectactsfunctionreturnfirstderivativspecificatoptionalunlesunabldeterminhappenconstantbuiltundefinparametspecificonverthaveoutotherwisproducproducdifferlinalgjacobianpackagexpressrequiresvectorcalculuvectorcalculusexpressionpackagebutvectoralsocomputdeterminantpartndonlyperformveccalalwayaccesslongexamplmakefuncttgrtablegopugvgoperatorgarrowgfvectorgcolumngjtevalfunctjtghmmatrixgpvagfbgfmfrhothetaphisincosnrhosgrhogthetagphigsingcosgjspi"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 257 31 "VecCalc[JacobianDeterminant] - " }{TEXT -1 66 "Calcul ates the Jacobian Determinant of a Coordinate Transformation" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The alias can be used a fter execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 258 31 " JDet = JacobianD eterminant" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }} {PARA 0 "" 0 "" {TEXT 256 87 " JacobianDeterminant(T, vars) JDet(T , vars) VecCalc[JacobianDeterminant](T, vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 12 " T - " }{TEXT -1 104 "a coordinate transformation in the fo rm of a list or Vector of n arrow-defined functions of n variables." } }{PARA 0 "" 0 "" {TEXT 260 12 " vars - " }{TEXT -1 1 "(" }{TEXT 261 8 "optional" }{TEXT -1 89 ") sequence, list or Vector of names to \+ be used as independent variables for the function." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 " " {TEXT -1 211 "The Jacobian determinant of a coordinate transformatio n is the determinant of the Jacobian matrix. Notice it does not includ e an absolute value as required for a coordinate transformation in a m ultiple integral." }}{PARA 15 "" 0 "" {TEXT 262 19 "JacobianDeterminan t" }{TEXT -1 92 " acts on a list or Vector of arrow-defined functions and returns an arrow-defined function." }}{PARA 15 "" 0 "" {TEXT -1 207 "The specification of variables is optional unless Maple is unable to determine the variables for the function. This can happen if the f unction is constant, is a built-in function or is an undefined functio n." }}{PARA 15 "" 0 "" {TEXT -1 64 "The Jacobian determinant can also \+ be computed using the command " }{HYPERLNK 17 "VectorCalculus[Jacobian ]" 2 "VectorCalculus[Jacobian]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorCalculus" 2 "VectorCalculus" "" }{TEXT -1 52 " package by in cluding the 'determinant'=true option." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 263 19 "JacobianDeterminant" }{TEXT -1 16 " \+ is part of the " }{TEXT 265 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 264 19 "JacobianDeterminant" }{TEXT -1 35 " only after performing the command " }{TEXT 266 13 "with(VecCal c)" }{TEXT -1 4 " or " }{TEXT 267 34 "with(VecCalc, JacobianDeterminan t)" }{TEXT -1 56 ". The function can always be accessed in the long f orm " }{TEXT 268 28 "VecCalc[JacobianDeterminant]" }{TEXT -1 1 "." }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(VecCalc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "T:=MakeFunction(, );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'RTABLEG6%\")o^P8-%'MATR IXG6#7$7#f*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF2,&*$)9$\"\"#\"\"\"F;* $)9%F:F;F;F2F2F27#f*F/F2F3F2*&F>F;F9F;F2F2F2&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "JT:=JacobianDeterminant(T); \+ JT(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#JTGf*6$%\"uG%\"vG6\"6$% )operatorG%&arrowGF),&*&\"\"#\"\"\")9$F/F0F0*&F/F0)9%F/F0!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\")%\"aGF%F&F&*&F%F&)% \"bGF%F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "S:=MF([rho ,theta,phi], [rho*sin(phi)*cos(theta),\nrho*sin(phi)*sin(theta), rho*c os(phi)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7%f*6%%$rhoG%&thet aG%$phiG6\"6$%)operatorG%&arrowGF+*(9$\"\"\"-%$sinG6#9&F1-%$cosG6#9%F1 F+F+F+f*F'F+F,F+*(F0F1F2F1-F3F8F1F+F+F+f*F'F+F,F+*&F0F1-F7F4F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "JS:=JDet(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#JSGf*6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG% &arrowGF*,$*&-%$sinG6#9&\"\"\")9$\"\"#F4!\"\"F*F*F*" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " JacobianMatrix" 2 "JacobianMatrix" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " VectorCalculus[Jacobian]" 2 "VectorCalculus[Jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[jacobian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[det]" 2 "linalg[det]" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 1 "," }{TEXT -1 1 " " }{HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" } {TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13375168 }{RTABLE M7R0 I5RTABLE_SAVE/13375168X*%)anythingG6"6"[gl!#%!!!"#"#f*6$%"uG%"vG6"6$%)operatorG %&arrowGF+,&*$9$""#"""*$9%F2F3F+F+F+f*F(F+F,F+*&F5F3F1F3F+F+F+F+ } of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " JacobianMatrix" 2 "JacobianMatrix" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " VectorCalculus[Jacobian]" 2 "VectorCalculus[Jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[jacobian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[det]" 2 "linalg[det]" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 1 "," }{TEXT -1 1 " " }{HYPERLNK 17 "Multipleint" 2 "Mult"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaltimesbulletitemfunctveccalcjacobiandeterminantcalculatesjacobiandeterminantcoordinattransformataliacanusedfterexecutvcaliacommandjdetjacobiandeterminantcallsequencvarsparameterformlistvectorarrowdefinfunctionvariabloptionalnameindependdescripttransformatiomatrixnoticdoesincludabsolutvalurequirultiplintegraljacobiandeterminanactsreturnspecificatunlesunabldeterminhappenunctconstantbuiltundefinfunctioalsocomputusingvectorcalculupackagcludtrueoptionpartformonlyafterperformwithveccalalwayaccesslongormexamplmakefuncttgrtablegmatrixgugvgoperatorgarrowgfvectorgcolumngjtjtgfagfbgfmfrhothetaphisincosnrhoossgrhogthetagphigsingcosgjsjsgfthetagcopyrightarthurbelmonthilipyasskindepartmmathematictexauniversitjacobianmatrixlinalgdetdiffopmultipleintcoordconversDot(Mathematics/Packages/VecCalc/Commands/&.Gradient%Mathematics/Packages/VecCalc/GradientGradient.Mathematics/Packages/VecCalc/Commands/GradientGradient)Mathematics/Packages/VecCalc/Aliases/GradHessian%Mathematics/Packages/VecCalc/Hessian Hessian.Mathematics/Packages/VecCalc/Commands/Hessian Hessian)Mathematics/Packages/VecCalc/Aliases/HessJacobianDeterminant0Mathematics/Packages/VecCalc/JacobianDeterminantJacobianDeterminant9Mathematics/Packages/VecCalc/Commands/JacobianDeterminantJacobianDeterminant)Mathematics/Packages/VecCalc/Aliases/JDetJacobianMatrix+Mathematics/Packages/VecCalc/JacobianMatrixJacobianMatrix4Mathematics/Packages/VecCalc/Commands/JacobianMatrixJacobianMatrix(Mathematics/Packages/VecCalc/Aliases/Jac Laplacian&Mathematics/Packages/VecCalc/Laplacian Laplacian/Mathematics/Packages/VecCalc/Commands/Laplacian Laplacian(Mathematics/Packages/VecCalc/Aliases/Lap!LeadingPrincipalMinorDeterminants>Mathematics/Packages/VecCalc/LeadingPrincipalMinorDeterminantsminant9Mathematics/Packages/VecCalc/Commands/JacobianDeterminantE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 27 "VecCalc[ScalarPotential] - " }{TEXT -1 81 "Calculates the Scalar Potential of a Vector Field in Arrow Notation If It Exists " }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The alias ca n be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias " "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 272 27 " SPot \+ = ScalarPotential" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences: " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 79 " ScalarPotential(F , vars) SPot(F, vars) VecCalc[ScalarPotential](F, vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 12 " F - " }{TEXT -1 91 "a vector field in the form o f a list or Vector of n arrow-defined functions of n variables." }} {PARA 0 "" 0 "" {TEXT 259 12 " vars - " }{TEXT -1 1 "(" }{TEXT 260 8 "optional" }{TEXT -1 89 ") sequence, list or Vector of names to \+ be used as independent variables for the function." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 77 "A scalar potential of a vector field F is a scalar field whose gradient is F." }}{PARA 15 "" 0 "" {TEXT 261 15 "ScalarPotential" }{TEXT -1 171 " acts on a list or Vector of arrow-defined functions and returns \+ its scalar potential as an arrow-defined function, if it exists. If t he scalar potential does not exist, " }{TEXT 262 15 "ScalarPotential" }{TEXT -1 21 " returns 'undefined'." }}{PARA 15 "" 0 "" {TEXT -1 212 " The specification of variables is optional unless Maple is unable to d etermine the variables for the vector field. This can happen if the fu nctions are constant, are built-in functions or are undefined function s." }}{PARA 15 "" 0 "" {TEXT 263 15 "ScalarPotential" }{TEXT -1 26 " d iffers from the command " }{HYPERLNK 17 "linalg[potential]" 2 "linalg[ potential]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg " "" }{TEXT -1 203 " package which acts on a list or vector of express ions and requires the specification of the variables. It returns true or false and requires an extra argument to returns the potential as a n expression." }}{PARA 15 "" 0 "" {TEXT 264 15 "ScalarPotential" } {TEXT -1 26 " differs from the command " }{HYPERLNK 17 "VectorCalculus [ScalarPotential]" 2 "VectorCalculus[ScalarPotential]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorCalculus" 2 "VectorCalculus" "" } {TEXT -1 208 " package which acts on a VectorField of expressions and \+ returns an expression for the scalar potential or NULL. The VectorCal culus package requires the specification of a coordinate system and it s variables." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 265 15 "ScalarPotential" }{TEXT -1 16 " is part of the " }{TEXT 267 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " } {TEXT 266 15 "ScalarPotential" }{TEXT -1 35 " only after performing th e command " }{TEXT 268 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 269 30 "with(VecCalc, ScalarPotential)" }{TEXT -1 56 ". The function \+ can always be accessed in the long form " }{TEXT 270 24 "VecCalc[Scala rPotential]" }{TEXT -1 13 ". The alias " }{TEXT 271 4 "SPot" }{TEXT -1 47 " can be used only after performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g:=MakeFunction([w,x,y,z],x^2+w*exp(y)*sin(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6&%\"wG%\"xG%\"yG%\"zG6\"6$%)o peratorG%&arrowGF+,&*$)9%\"\"#\"\"\"F4*(9$F4-%$expG6#9&F4-%$sinG6#9'F4 F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "G:=Grad(g);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'RTABLEG6%\")G\"yL\"-%'VECTORG 6#7&f*6&%\"wG%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF3*&-%$expG6#9&\" \"\"-%$sinG6#9'F " 0 "" {MPLTEXT 1 0 19 "ScalarPotential(G);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6&%\"wG%\"xG%\"yG%\"zG6\"6$%)operat orG%&arrowGF),&*$)9%\"\"#\"\"\"F2*(9$F2-%$expG6#9&F2-%$sinG6#9'F2F2F)F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ScalarPotential(G,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6&%\"aG%\"bG%\"cG%\"dG6 \"6$%)operatorG%&arrowGF),&*$)9%\"\"#\"\"\"F2*(9$F2-%$expG6#9&F2-%$sin G6#9'F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "F:=MF([x ,y,z],<2*y| exp(y)*sin(z)| exp(y)*cos(z)>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'RTABLEG6%\")G$yL\"-%'VECTORG6#7%f*6%%\"xG%\"yG %\"zG6\"6$%)operatorG%&arrowGF2,$*&\"\"#\"\"\"9%F9F9F2F2F2f*F.F2F3F2*& -%$expG6#F:F9-%$sinG6#9&F9F2F2F2f*F.F2F3F2*&F=F9-%$cosGFBF9F2F2F2&%'Ve ctorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ScalarPote ntial(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " linalg[potential]" 2 "linalg[potential]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "VectorCalculus[ScalarPotential]" 2 "VectorCalculus[Scala rPotential]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "Gradient" 2 "Gradient" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curl" 2 "Curl" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VectorPotential" 2 "VectorPotential" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13378128 13378328 }{RTABLE M7R0 I5RTABLE_SAVE/13378128X*%)anythingG6"6"[gl!$%!!!"%"%f*6&%"wG%"xG%"yG%"zG6"6$%)o peratorG%&arrowGF-*&-%$expG6#9&"""-%$sinG6#9'F6F-F-F-f*F(F-F.F-,$9%""#F-F-F-f*F (F-F.F-*(9$F6F2F6F7F6F-F-F-f*F(F-F.F-*(FAF6F2F6-%$cosGF9F6F-F-F-F- } {RTABLE M7R0 I5RTABLE_SAVE/13378328X*%)anythingG6"6"[gl!$%!!!"$"$f*6%%"xG%"yG%"zG6"6$%)opera torG%&arrowGF,,$9%""#F,F,F,f*F(F,F-F,*&-%$expG6#F1"""-%$sinG6#9&F8F,F,F,f*F(F,F -F,*&F5F8-%$cosGF;F8F,F,F,F, } "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " linalg[potential]" 2 "linalg[potential"ibmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormaloutputbulletitemfunctveccalcscalarpotentialcalculatscalarpotentialvectorfieldarrownotatexistaliacausedafterexecutvcaliacommandspotcallsequencvarsparameterformlistdefinfunctionvariabloptionalnameindependdescriptwhosgradiactsreturnhedoesundefinspecificatunlesunabletermincanhappenfunctionconstantbuiltifferlinalgpackagexpresionsrequirtruefalsextraargumexpressdiffervectorcalculuvectorfieldnullvectorcalculucoordinatsystempartonlyperformthwithalwayaccesslongscalarpotentialexamplmakefunctexpsinggfwgxgygzgperatorgarrowgfexpgsinggradggrtablegylvectorgoperatorgfhfcosgfrowoperatorgagbgcgdgmfcosfgcosgfbfvectorgrowgscalarpotntialundefinedgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopcurlvectorpotential7Mathematics/Packages/VecCalc/Commands/OutputVectorTypeOutputTypes6Mathematics/Packages/VecCalc/Commands/ScalarPotentialScalarPotential2Mathematics/Packages/VecCalc/Commands/SurfaceArea Surface4Mathematics/Packages/VecCalc/Commands/SurfaceForgetSurfaceForget7Mathematics/Packages/VecCalc/Commands/SurfaceIntScalarSurfaceIntScalar<Mathematics/Packages/VecCalc/Commands/SurfaceIntScalarInertSurfaceIntScalar7Mathematics/Packages/VecCalc/Commands/SurfaceIntVectorSurfaceIntVector<Mathematics/Packages/VecCalc/Commands/SurfaceIntVectorInertSurfaceIntVector4Mathematics/Packages/VecCalc/Commands/SurfaceNormal Surface:Mathematics/Packages/VecCalc/Commands/SurfaceNormalLength Surface6Mathematics/Packages/VecCalc/Commands/SurfaceTangents Surface.Mathematics/Packages/VecCalc/Commands/VCalias VCalias6Mathematics/Packages/VecCalc/Commands/VectorPotentialVectorPotential/Mathematics/Packages/VecCalc/Commands/cyl2rectCoordConversion3D.Mathematics/Packages/VecCalc/Commands/cyl2sphCoordConversion3D.Mathematics/Packages/VecCalc/Commands/deg2radAngleConversion+Mathematics/Packages/VecCalc/Commands/diffMappedFunctions3Mathematics/Packages/VecCalc/Commands/evalFunctionevalFunction*Mathematics/Packages/VecCalc/Commands/intMappedFunctionsSurface$Mathematics/Packages/VecCalc/SurfaceSurface5Mathematics/Packages/VecCalc/Commands/SurfaceTangentsSurface3Mathematics/Packages/VecCalc/Commands/SurfaceNormalSurface9Mathematics/Packages/VecCalc/Commands/SurfaceNormalLengthSurface1Mathematics/Packages/VecCalc/Commands/SurfaceAreaSurface'Mathematics/Packages/VecCalc/Aliases/STSurface'Mathematics/Packages/VecCalc/Aliases/SNSurface(Mathematics/Packages/VecCalc/Aliases/SNLSurface'Mathematics/Packages/VecCalc/Aliases/SAE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Cour ier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE "" -1 -1 "C ourier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 27 "VecCalc[VectorPotential] - " }{TEXT -1 81 "Calculates the Vector Potential of a Vector Field in Arrow Notation If It Exists " }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The alias ca n be used after execution of the " }{HYPERLNK 17 "VCalias" 2 "VCalias " "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 259 27 " VPot \+ = VectorPotential" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences: " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 106 " VectorPotential( F, vars, outtype) VPot(F, vars, outtype) VecCalc[VectorPotential]( F, vars, outtype)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 13 " F - " }{TEXT -1 91 " a vector field in the form of a list or Vector of 3 arrow-defined func tions of 3 variables " }}{PARA 0 "" 0 "" {TEXT 260 13 " vars - " }{TEXT -1 1 "(" }{TEXT 261 8 "optional" }{TEXT -1 91 ") sequence, list or Vector of 3 names to be used as independent variables for the func tion." }}{PARA 0 "" 0 "" {TEXT 275 13 " outtype - " }{TEXT 276 24 "( optional) output type: " }{TEXT 277 51 "'list', 'Vector', 'Vector[row] ' or 'Vector[column]'" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descrip tion:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 37 "A vector potent ial of a vector field " }{TEXT 264 1 "F" }{TEXT -1 33 " is a vector fi eld whose curl is " }{TEXT 265 1 "F" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT 263 15 "VectorPotential" }{TEXT -1 193 " acts on a list or Vec tor of 3 arrow-defined functions and returns its vector potential as a list or Vector of 3 arrow-defined functions, if it exists. If the ve ctor potential does not exist, " }{TEXT 262 15 "VectorPotential" } {TEXT -1 21 " returns 'undefined'." }}{PARA 15 "" 0 "" {TEXT -1 212 "T he specification of variables is optional unless Maple is unable to de termine the variables for the vector field. This can happen if the fun ctions are constant, are built-in functions or are undefined functions ." }}{PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 278 7 "outtype" } {TEXT -1 84 " parameter is specified, then the output is converted to \+ have the type specified by " }{TEXT 279 7 "outtype" }{TEXT -1 65 ". Ot herwise the output type matches the type of the vector field " }{TEXT 280 1 "F" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT 266 15 "VectorPoten tial" }{TEXT -1 26 " differs from the command " }{HYPERLNK 17 "linalg[ vecpotent]" 2 "linalg[vecpotent]" "" }{TEXT -1 8 " in the " } {HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 213 " package which acts on a list or vector of expressions and requires the specification of \+ the variables. It returns true or false and requires an extra argumen t to returns the potential as a vector of expressions." }}{PARA 15 "" 0 "" {TEXT 267 15 "VectorPotential" }{TEXT -1 26 " differs from the co mmand " }{HYPERLNK 17 "VectorCalculus[VectorPotential]" 2 "VectorCalcu lus[VectorPotential]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorC alculus" 2 "VectorCalculus" "" }{TEXT -1 223 " package which acts on a VectorField of expressions and returns a VectorField of expressions f or the vector potential or NULL. The VectorCalculus package requires \+ the specification of a coordinate system and its variables." }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 268 15 "VectorPotential " }{TEXT -1 16 " is part of the " }{TEXT 270 7 "VecCalc" }{TEXT -1 41 " package, and so can be used in the form " }{TEXT 269 15 "VectorPoten tial" }{TEXT -1 35 " only after performing the command " }{TEXT 271 13 "with(VecCalc)" }{TEXT -1 4 " or " }{TEXT 272 30 "with(VecCalc, Vec torPotential)" }{TEXT -1 56 ". The function can always be accessed in the long form " }{TEXT 273 24 "VecCalc[VectorPotential]" }{TEXT -1 13 ". The alias " }{TEXT 274 4 "VPot" }{TEXT -1 47 " can be used only after performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" " " }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCal c): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A:=MakeFunc tion([x,y,z],[x+y+z, x*y*z, x*y+y*z+z*x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,( 9$\"\"\"9%F19&F1F+F+F+f*F'F+F,F+*(F0F1F2F1F3F1F+F+F+f*F'F+F,F+,(*&F2F1 F0F1F1*&F2F1F3F1F1*&F3F1F0F1F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C:=Curl(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG 7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(9$\"\"\"9&F1*&9%F1F0F 1!\"\"F+F+F+f*F'F+F,F+,(F1F1F4F5F2F5F+F+F+f*F'F+F,F+,&*&F4F1F2F1F1F1F5 F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "VectorPotential(C );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%f*6%%\"xG%\"yG%\"zG6\"6$%)oper atorG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*&#F/\"\"#F/*$)F.F5F/F/F2F1F/F )F)F)f*F%F)F*F),(*&F.F/9$F/F2*&#F/F5F/F6F/F2*(F;F/F1F/F.F/F/F)F)F)\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "C-Curl(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "VectorPotential(C,[a,b,c],'Vector');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")%y[P\"-%'MATRIXG6#7%7#f*6%%\"aG%\"b G%\"cG6\"6$%)operatorG%&arrowGF1,*9&\"\"\"*&9%F7F6F7!\"\"*&#F7\"\"#F7* $)F6F=F7F7F:F9F7F1F1F17#f*F-F1F2F1,(*&F6F79$F7F:*&#F7F=F7F>F7F:*(FDF7F 9F7F6F7F7F1F1F17#\"\"!&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "VectorPotential(A,Vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 " - Copyright 1995-2003 by Arthur Belmonte and Philip B. Yasskin\n \+ Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "VecCalc" 2 "VecC alc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[vecpotent]" 2 "linalg[v ecpotent]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VectorCalculus[VectorPot ential]" 2 "VectorCalculus[VectorPotential]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " Curl" 2 "Curl" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Divergence" 2 "Diver gence" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "ScalarPotential" 2 "ScalarPo tential" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13748784 } {RTABLE M7R0 I5RTABLE_SAVE/13748784X*%)anythingG6"6"[gl!#%!!!"$"$f*6%%"aG%"bG%"cG6"6$%)opera torG%&arrowGF,,*9&"""*&9%F2F1F2!""*$F1""##F5F7F4F2F,F,F,f*F(F,F-F,,(*&F1F29$F2F 5F6F8*(F " 0 "" {MPLTEXT 1 0 23 "with (VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Muin t(x^4*y^3*z^2,x=1..2,y=3..4,z=5..6); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*()%\"xG\"\"%\"\"\")%\"yG\"\"$F.)%\" zG\"\"#F./F,;F.F4/F0;F1F-/F3;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "muint (x^4*y^3*z^2,x=1..2,y=3..4,z=5..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "muint(x^4* y^3*z^2,x=1..2,y=3..4,z=5..6,'step'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*()%\"xG\"\"%\"\"\")%\"yG\"\"$F.)%\"zG\"\"#F./F, ;F.F4/F0;F1F-/F3;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G- %$IntG6$-F&6$-%%EvalG6$,$**\"\"&!\"\"%\"xGF/%\"yG\"\"$%\"zG\"\"#\"\"\" /F1;F6F5/F2;F3\"\"%/F4;F/\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" ~G-%$IntG6$-F&6$,$**\"#J\"\"\"\"\"&!\"\"%\"yG\"\"$%\"zG\"\"#F-/F0;F1\" \"%/F2;F.\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$-%%Ev alG6$,$**\"#J\"\"\"\"#?!\"\"%\"yG\"\"%%\"zG\"\"#F./F1;\"\"$F2/F3;\"\"& \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$,$*(\"%&3\"\" \"\"\"\"%!\"\"%\"zG\"\"#F+/F.;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%%EvalG6$,$*(\"%&3\"\"\"\"\"#7!\"\"%\"zG\"\"$F+/F.;\"\"& \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"&N()*\"#7" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur B elmonte and Philip B. Yasskin\n Department of Mathematics, Texas \+ A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "student[Doubleint]" 2 "student[Doublein t]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "student[Tripleint]" 2 "student[ Tripleint]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LineIntScalar " 2 "LineIntScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LineIntVector" 2 "LineIntVector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntScalar " 2 "SurfaceIntScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntVe ctor" 2 "SurfaceIntVector" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } 1 "" {XPPMATH 20 "6#/%\"~G#\"&N()*\"#7" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur B elmonte and Philip B. Yasskin\n Department of Mathematics, Texas \+ A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "student[Doubleint]" 2 "student[Doublein t]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "student[Tripleint]" 2 "student[ Tripleint]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JacobianDeterminant" 2 "JacobianDeterminant" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LineIntScalar " 2 "LineIntScalar"help=" """""""""""""""""""""""""""""""""""here ""herwis"hes"hese"hess """""hessg"hessi"hessian'7"""""" """hf ""hfg ""hg ""hgf"""hgg"hi"higf"high"hile"hilip"hink"his"hm" horizontal ""howev """""""hyper" hyperlink$""""""""""""""""""""""""""""""""""""iabl"""ial"ian"ias"ibm$""""""""""""""""""""""""""""""""""""sur ""surf""""surfa ""surfac/C"""""""""" " surfacear" surfacearea""" surfaceforget"""" surfacei" surfaceint"surfaceintscal ""surfaceintscalar"""""""surfaceintscalarinert "" surfaceintv"surfaceintvect ""surfaceintvector"""""""surfaceintvectorinert "" surfacenor" surfacenormal""""eintvectorbmintelntmaplinputcourimathtimehyperlinkoutputhelpheadcouriernormalbulletitemfixedwidthfunctionveccalcmultipleintdisplayinertmulitiplintegralveccalccomputmultiplbothcommandcandisplaintermediatstepsaliasaliasesusedafterexecutvcaliacaliamuintcallsequencxnmultipleinstepintparameterexpressintegrandeachnamerangspecifvariablintegratoptionalrangerequirparametindicatdescriptfirstargumfollowargumentntegratincludnumericalappearordertheyevaluatdifferentialusingvaluevalfcalculatwithoutwhilallmultipleintreturnyoudoneedquotaroundassignfunctionalitstuddoubleinttripleinttripleintpackagbutalsoworkhighdimensionalintegralsthespartonlyperformwithfunctalwayaccesslongformsedvcaliasexamplesmuinintgxgygzgevalgxgfevalgcopyrightarthurelmontphilipyasskindepartmmathematictexauniversitdoubleinjacobiandeterminantlineintscalarlineintvectorsurfaceintscalarsurfaceintvctorsurfac",VecCalc,CforgetVecCalc,CurveForgetCforget"VecCalc,SAVecCalc,SNLVecCalc,SNVecCalc,STVecCalc,SurfaceAreaVecCalc,SurfaceNormalLengthVecCalc,SurfaceNormalVecCalc,SurfaceTangentsSASNLSNSTSurfaceAreaSurfaceNormalLengthSurfaceNormalSurfaceTangentssurface".VecCalc,SforgetVecCalc,SurfaceForgetSforget"DifferentialOperatorsdiffops"MaxMin"1VecCalc,GradVecCalc,GradientgradGradgradient"/VecCalc,HessVecCalc,HessianhessHesshessian(Mathematics/Packages/VecCalc/Aliases/SN Surface)Mathematics/Packages/VecCalc/Aliases/SNL Surface*Mathematics/Packages/VecCalc/Aliases/SPotScalarPotential(Mathematics/Packages/VecCalc/Aliases/ST Surface-Mathematics/Packages/VecCalc/Aliases/SforgetSurfaceForget)Mathematics/Packages/VecCalc/Aliases/SisSurfaceIntScalar)Mathematics/Packages/VecCalc/Aliases/SivSurfaceIntVector*Mathematics/Packages/VecCalc/Aliases/VPotVectorPotentialHMathematics/Packages/VecCalc/Commands/LeadingPrincipalMinorDeterminants&LeadingPrincipalMinorDeterminants-Mathematics/Packages/VecCalc/Commands/Length Length,Mathematics/Packages/VecCalc/Commands/LimitMappedFunctions4Mathematics/Packages/VecCalc/Commands/LineIntScalarLineIntScalar9Mathematics/Packages/VecCalc/Commands/LineIntScalarInertLineIntScalar4Mathematics/Packages/VecCalc/Commands/LineIntVectorLineIntVector9Mathematics/Packages/VecCalc/Commands/LineIntVectorInertLineIntVector3Mathematics/Packages/VecCalc/Commands/MakeFunctionMakeFunction2Mathematics/Packages/VecCalc/Commands/MultipleintMultipleint7Mathematics/Packages/VecCalc/Commands/OutputMatrixTypeOutputTypes} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Fixed Width" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 33 " VecCalc[LineIntScalarInert] - " }{TEXT -1 43 "D isplays a Line Integral of a Scalar Field " }}{PARA 0 "" 0 "" {TEXT 257 33 " VecCalc[LineIntScalar] - " }{TEXT -1 42 "Computes a Li ne Integral of a Scalar Field" }}{PARA 0 "" 0 "" {TEXT -1 45 "Both com mands can display intermediate steps." }}{PARA 0 "" 0 "" {TEXT 26 8 "A liases:" }{TEXT -1 50 " - The aliases can be used after execution of t he " }{HYPERLNK 17 "VCalias" 2 "vc_aliases" "" }{TEXT -1 9 " command. " }}{PARA 257 "" 0 "" {TEXT -1 11 " Lis = " }{TEXT 263 18 "LineInt ScalarInert" }}{PARA 257 "" 0 "" {TEXT -1 11 " lis = " }{TEXT 264 13 "LineIntScalar" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 265 18 "Li neIntScalarInert" }{TEXT -1 20 "(f,r,var=rng) " }}{PARA 256 "" 0 "" {TEXT -1 30 " Lis(...) VecCalc[" }{TEXT 271 18 "LineI ntScalarInert" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 266 18 "LineIntScalarInert" }{TEXT -1 20 "(f,r,var=rng,'step ')" }}{PARA 256 "" 0 "" {TEXT -1 30 " Lis(...,'step') VecCalc[" } {TEXT 272 18 "LineIntScalarInert" }{TEXT -1 13 "](...,'step')" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 267 13 "LineIntScalar" } {TEXT -1 25 "(f,r,var=rng) " }}{PARA 256 "" 0 "" {TEXT -1 30 " lis(...) VecCalc[" }{TEXT 269 13 "LineIntScalar" } {TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 268 13 "LineIntScalar" }{TEXT -1 25 "(f,r,var=rng,'step') " }}{PARA 256 "" 0 "" {TEXT -1 30 " lis(...,'step') VecCalc[" }{TEXT 270 13 "LineIntScalar" }{TEXT -1 13 "](...,'step')" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 258 12 " f - " }{TEXT -1 51 "a scalar function of n variables in arrow notation." }}{PARA 0 "" 0 "" {TEXT 259 12 " r - " }{TEXT -1 76 "a curve in the form of a list of n arrow-defined functions of one parameter." }} {PARA 0 "" 0 "" {TEXT 261 12 " var - " }{TEXT -1 48 "the variable of integration and curve parameter." }}{PARA 0 "" 0 "" {TEXT 262 12 " rng - " }{TEXT -1 45 "the range of the parameter to integrate ov er." }}{PARA 0 "" 0 "" {TEXT 260 12 " 'step' - " }{TEXT -1 81 "(opti onal) parameter to indicate that the intermediate steps should be disp layed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 273 18 "LineIntScalarInert" }{TEXT -1 237 " displays the line integral of a scalar field, where the first argument is the scalar field, the second argument is the curve and th e third argument is the variable and range of integration. The integr al can then be evaluated using the " }{TEXT 274 5 "value" }{TEXT -1 5 " or " }{TEXT 275 5 "evalf" }{TEXT -1 9 " command." }}{PARA 15 "" 0 " " {TEXT 276 13 "LineIntScalar" }{TEXT -1 86 " calculates the line inte gral of a scalar field without first displaying the integral." }} {PARA 15 "" 0 "" {TEXT -1 7 "If the " }{TEXT 277 6 "'step'" }{TEXT -1 144 " parameter is included, Maple displays the line integral of a sca lar field and then calculates it while displaying all the intermediate steps. " }{TEXT 279 18 "LineIntScalarInert" }{TEXT -1 48 " then retu rns the inert multiple integral while " }{TEXT 280 13 "LineIntScalar" }{TEXT -1 55 " returns its value. You do not need the quotes around \+ " }{TEXT 278 6 "'step'" }{TEXT -1 52 " if the variable step has not be en assigned a value." }}{PARA 15 "" 0 "" {TEXT 285 13 "LineIntScalar" }{TEXT -1 27 " is similar to the command " }{HYPERLNK 17 "VectorCalcul us[PathInt]" 2 "VectorCalculus[PathInt]" "" }{TEXT -1 8 " in the " } {HYPERLNK 17 "VectorCalculus" 2 "VectorCalculus" "" }{TEXT -1 37 " pac kage with the domain chosen as a " }{TEXT 286 4 "Path" }{TEXT -1 138 " . However, that command uses expressions instead of arrow-defined fun ctions, cannot display the integral nor show the intermediate steps." }}{PARA 15 "" 0 "" {TEXT -1 32 "These functions are part of the " } {TEXT 281 7 "VecCalc" }{TEXT -1 71 " package, and so can be used by na me only after performing the command " }{TEXT 282 13 "with(VecCalc)" } {TEXT -1 4 " or " }{TEXT 283 22 "with(VecCalc,function)" }{TEXT -1 58 ". The functions can always be accessed in the long forms " }{TEXT 284 17 "VecCalc[function]" }{TEXT -1 61 ". The aliases can be used on ly after performing the command " }{HYPERLNK 17 "VCalias" 2 "VCalias" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCa lc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=MakeFun ction([x,y,z],2/9*y*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%% \"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,$*&#\"\"#\"\"*\"\"\"*&9%F39&F 3F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r:=MakeFunct ion(t,[2*t,3*sin(t),3*cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG7%f*6#%\"tG6\"6$%)operatorG%&arrowGF),$*&\"\"#\"\"\"9$F0F0F)F)F)f* F'F)F*F),$*&\"\"$F0-%$sinG6#F1F0F0F)F)F)f*F'F)F*F),$*&F5F0-%$cosGF8F0F 0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "LineIntScalarIne rt(f,r,t=0..Pi/2); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$,$**\"\"#\"\"\"-%$sinG6#%\"tGF)-%$cosGF,F)\"#8#F)F(F)/F-;\"\"!,$*&F (!\"\"%#PiGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#8#\"\"\"\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "LineIntScalar(f,r,t=0.. Pi/2,'step');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$**\"\"#\" \"\"-%$sinG6#%\"tGF)-%$cosGF,F)\"#8#F)F(F)/F-;\"\"!,$*&F(!\"\"%#PiGF)F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%%EvalG6$,$*&\"#8#\"\"\"\" \"#)-%$cosG6#%\"tGF-F,!\"\"/F2;\"\"!,$*&F-F3%#PiGF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*$\"#8#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#8#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "VectorCalculus[PathInt]( 2/9*y*z, [x,y,z] = Path( <2* t,3*sin(t),3*cos(t)>, t=0..Pi/2 ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*$\"#8#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:= MF([x,y],3*x*sin(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF),$*(\"\"$\"\"\"9$F0-%$sinG6#9%F0F0F)F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R:=MF(t,[ln(t),2*(-t )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$%#lnGf*6#%\"tG6\"6$%)o peratorG%&arrowGF*,$*&\"\"#\"\"\"9$F1!\"\"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Lis(g,R,t=1..2); value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$,$*,\"\"$\"\"\"-%#lnG6#%\"tGF)-%$sinG6#,$*& \"\"#F)F-F)F)F),&*&\"\"%F))F-F3F)F)F)F)#F)F3-%$absGF,!\"\"F;/F-;F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$,$*,\"\"$\"\"\"-%#lnG6#%\"t GF)-%$sinG6#,$*&\"\"#F)F-F)F)F),&*&\"\"%F))F-F3F)F)F)F)#F)F3-%$absGF,! \"\"F;/F-;F)F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#[:[K$!#5" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "h:=MF([x,y,z,w],x*z*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6&%\"xG%\"yG%\"zG%\"wG6\"6$%)operatorG%&arrowGF +*(9$\"\"\"9&F19'F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:=MF(t,[2*t,t^2,t^2,2*t^3/3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"rG7&f*6#%\"tG6\"6$%)operatorG%&arrowGF),$*&\"\"#\"\"\"9$F0F0F)F)F )f*F'F)F*F)*$)F1F/F0F)F)F)F2f*F'F)F*F),$*&#F/\"\"$F0*$)F1F9F0F0F0F)F)F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "lis(h,r,t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"$G\"\"$*=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lis(h,r,t=0..1,'step'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$**\"\"%\"\"\"\"\"$!\"\"%\"tG\"\"',&\"\"#F)*& F/F))F,F/F)F)F)F)/F,;\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G -%%EvalG6$,&*&#\"\")\"#F\"\"\"*$)%\"tG\"\"*F-F-F-*&#F+\"#@F-*$)F0\"\"( F-F-F-/F0;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"$G\"\"$* =" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by A rthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " MakeFunction" 2 "MakeFunction" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curv e" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "LineIntVector" 2 "LineIntVector" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "SurfaceIntScalar" 2 "SurfaceIntScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntVector" 2 "SurfaceIntVector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VectorCalculus[PathInt]" 2 "VectorCalculus[PathI nt]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } \"( F-F-F-/F0;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"$G\"\"$* =" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by A rthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "VecCalc" 2 "VecCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " MakeFunction" 2 "MakeFunction" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curv e" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPEne ""near"ned"need' """""""""negat ""neintscalarinert" neintvector"neintvectorinert"nest"""neutral""""new"ng"""""nge" ngleconvers"ngprincipalminordeterminant ""ngth"nly"nmatrix"no ""nochk"non ""nor"""""nordeterminant"norm " ""normal4""""""""""""""""""""""""""""""""""""notat/ """""""""""note ""notic ""nrho ""ns""""nstant"nstead"nt*""""""""""""""""""""""""""""""""""""partial"""path ""pathi"pathint"pe" pecificat"peratorg"""perf"perfo"perform2""""""""""""""""""""""""""""""" performin""""period" perpendicular"pf"pg"""pgf ""phi!"""""livg"ln ""lng""""lngf"load"local ""longs""""""""""""""""""""""""""""low"lower"lp"lpm"lpmd """""lpmdg"lt" ltipleint"lue"lus ""lve"ly ""ma ""mai"mak"make""""rr"rray"rrg"rs"""rt ""rta"rtab"rtabl?F"""" """" "" """""rtableg{T""""" """"""""" """"""""""""""""rtg ""rthur"rule"rve "" rveforget"sa"""saddl"sag" bndcritpt" bndcritptsg"bo"both"""""bottom"boundar"boxed"brook"bs"bui"built#""""""""bul"bullet#"""""""""""""""""""""""""""""""""""but""""bviou"ca """"""JacobianDeterminant,VecCalc"JacobianMatrix,VecCalc" LPMD,VecCalc" Lap,VecCalc"Laplacian,VecCalc")LeadingPrincipalMinorDeterminants,VecCalc"Length,VecCalc" Limit,VecCalc"LineIntScalar,VecCalc"LineIntScalarInert,VecCalc"LineIntVector,VecCalc"LineIntVectorInert,VecCalc" Lis,VecCalc" Liv,VecCalc" MF,VecCalc"MakeFunction,VecCalc" Muint,VecCalc"Multipleint,VecCalc"OutputMatrixType,VecCalc"OutputVectorType,VecCalc"fintgevalgcosgmfggflnlngfperatorglngabsgfgfhgfwgcopyrightrthurbelmontphilipyasskindepartmmathematictexauniversitalsomultipleintlineintvectorsurfaceintscalarsurfaceintvectorpathircomputlinebothcommandcandisplaintermediatstepliasaliasusedafterexecuthevcaliavccommandlislineintscalarinertcallsequencneintscalarinertvarrnglineintscalarinertparameterfunctvariablarrownotatcurvformlistdefinparametintegratrangoveroptionalindicatdisplayeddescriptdisplayfirstargumsecondththirdintegralevaluatusingvaluevalfcalculatintegralwithoutincludscalarwhilallreturnsinertmultiplreturnyoudoneedquotaroundenassignsimilarvectorcalculuspathintvectorcalculupackagewithdomainchosenpathhowevusesexpressinsteadfunctioncannotnorshowthespartpackagnameonlyperformalwayaccesslonglyexamplveccalcmakefunfgfxgygzgoperatorgarrowgfmakefunctionsincosrgtgsingcosgflineintscalarinrtpiinttgfpig"},{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Fixed Width" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 33 " VecCalc[LineIntVectorInert] - " }{TEXT -1 43 "D isplays a Line Integral of a Vector Field " }}{PARA 0 "" 0 "" {TEXT 257 33 " VecCalc[LineIntVector] - " }{TEXT -1 42 "Computes a Li ne Integral of a Vector Field" }}{PARA 0 "" 0 "" {TEXT -1 45 "Both com mands can display intermediate steps." }}{PARA 0 "" 0 "" {TEXT 26 8 "A liases:" }{TEXT -1 50 " - The aliases can be used after execution of t he " }{HYPERLNK 17 "VCalias" 2 "vc_aliases" "" }{TEXT -1 9 " command. " }}{PARA 257 "" 0 "" {TEXT -1 11 " Liv = " }{TEXT 263 18 "LineInt VectorInert" }}{PARA 257 "" 0 "" {TEXT -1 11 " liv = " }{TEXT 264 13 "LineIntVector" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 265 18 "Li neIntVectorInert" }{TEXT -1 20 "(F,r,var=rng) " }}{PARA 256 "" 0 "" {TEXT -1 30 " Lis(...) VecCalc[" }{TEXT 271 18 "LineI ntVectorInert" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 266 18 "LineIntVectorInert" }{TEXT -1 20 "(F,r,var=rng,'step ')" }}{PARA 256 "" 0 "" {TEXT -1 30 " Lis(...,'step') VecCalc[" } {TEXT 272 18 "LineIntVectorInert" }{TEXT -1 13 "](...,'step')" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 267 13 "LineIntVector" } {TEXT -1 25 "(F,r,var=rng) " }}{PARA 256 "" 0 "" {TEXT -1 30 " lis(...) VecCalc[" }{TEXT 269 13 "LineIntVector" } {TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 268 13 "LineIntVector" }{TEXT -1 25 "(F,r,var=rng,'step') " }}{PARA 256 "" 0 "" {TEXT -1 30 " lis(...,'step') VecCalc[" }{TEXT 270 13 "LineIntVector" }{TEXT -1 13 "](...,'step')" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 258 12 " F - " }{TEXT -1 85 "a vector field in the form of a list of n functions o f n variables in arrow notation." }}{PARA 0 "" 0 "" {TEXT 259 12 " r - " }{TEXT -1 76 "a curve in the form of a list of n arrow-defin ed functions of one parameter." }}{PARA 0 "" 0 "" {TEXT 262 12 " var - " }{TEXT -1 48 "the variable of integration and curve parameter. " }}{PARA 0 "" 0 "" {TEXT 261 12 " rng - " }{TEXT -1 45 "the rang e of the parameter to integrate over." }}{PARA 0 "" 0 "" {TEXT 260 12 " 'step' - " }{TEXT -1 81 "(optional) parameter to indicate that the intermediate steps should be displayed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 273 18 "LineIntVectorInert" }{TEXT -1 237 " displays the line integral of a vector field, where the first argument is the vector field, the \+ second argument is the curve and the third argument is the variable an d range of integration. The integral can then be evaluated using the \+ " }{TEXT 274 5 "value" }{TEXT -1 5 " or " }{TEXT 275 5 "evalf" } {TEXT -1 9 " command." }}{PARA 15 "" 0 "" {TEXT 276 13 "LineIntVector " }{TEXT -1 86 " calculates the line integral of a vector field withou t first displaying the integral." }}{PARA 15 "" 0 "" {TEXT -1 7 "If th e " }{TEXT 277 6 "'step'" }{TEXT -1 144 " parameter is included, Maple displays the line integral of a vector field and then calculates it w hile displaying all the intermediate steps. " }{TEXT 279 18 "LineIntV ectorInert" }{TEXT -1 48 " then returns the inert multiple integral wh ile " }{TEXT 280 13 "LineIntVector" }{TEXT -1 55 " returns its value. \+ You do not need the quotes around " }{TEXT 278 6 "'step'" }{TEXT -1 52 " if the variable step has not been assigned a value." }}{PARA 15 " " 0 "" {TEXT 285 13 "LineIntVector" }{TEXT -1 27 " is similar to the c ommand " }{HYPERLNK 17 "VectorCalculus[LineInt]" 2 "VectorCalculus[Lin eInt]" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "VectorCalculus" 2 "Vec torCalculus" "" }{TEXT -1 37 " package with the domain chosen as a " } {TEXT 286 4 "Path" }{TEXT -1 192 ". However, that command uses expres sions instead of arrow-defined functions, cannot display the integral \+ nor show the intermediate steps and requires the specification of a co ordinate system." }}{PARA 15 "" 0 "" {TEXT -1 32 "These functions are \+ part of the " }{TEXT 281 7 "VecCalc" }{TEXT -1 71 " package, and so ca n be used by name only after performing the command " }{TEXT 282 13 "w ith(VecCalc)" }{TEXT -1 4 " or " }{TEXT 283 22 "with(VecCalc,function) " }{TEXT -1 58 ". The functions can always be accessed in the long fo rms " }{TEXT 284 17 "VecCalc[function]" }{TEXT -1 61 ". The aliases c an be used only after performing the command " }{HYPERLNK 17 "VCalias " 2 "VCalias" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(VecCalc): VCalias:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "F:=MakeFunction([x,y,z],[143*x^2*y,-71.5*y*z,4.2*x*z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG% &arrowGF+,$*(\"$V\"\"\"\")9$\"\"#F29%F2F2F+F+F+f*F'F+F,F+,$*($\"$:(!\" \"F2F6F29&F2F " 0 "" {MPLTEXT 1 0 37 "r:=MakeFunction(t,[2*t^3,3*t ^4,t^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%f*6#%\"tG6\"6$%)o peratorG%&arrowGF),$*&\"\"#\"\"\")9$\"\"$F0F0F)F)F)f*F'F)F*F),$*&F3F0) F2\"\"%F0F0F)F)F)f*F'F)F*F)*$)F2F/F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "LineIntVectorInert(F,r,t=0..1); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*&$\"&'H5\"\"!\"\"\")%\"tG\"#7F+F +*&$\"%uDF*F+)F-\"\"*F+!\"\"*&$\"++++!o\"!\")F+)F-\"\"'F+F+/F-;F*F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$P&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "LineIntVector(F,r,t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$P&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "G:=MF([x,y],);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG- %'RTABLEG6%\")/#3E\"-%'MATRIXG6#7$7#f*6$%\"xG%\"yG6\"6$%)operatorG%&ar rowGF2*$)9$\"\"$\"\"\"F2F2F27#f*F/F2F3F2*$)F8\"\"&F:F2F2F2&%'VectorG6# %'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "R:=MF(t,);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'RTABLEG6% \")%[3E\"-%'MATRIXG6#7$7#f*6#%\"tG6\"6$%)operatorG%&arrowGF1*$)-%$cosG 6#9$\"\"$\"\"\"F1F1F17#f*F/F1F2F1,$*&-%$sinGF9F<,&F " 0 "" {MPLTEXT 1 0 29 "Liv(G,R,t=0..Pi/2); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"$\"\"\")-%$cosG6#%\"tG\"#6F),(-%$sinGF -F)*$)F+\"\"&F)!\"\"*$)F+\"\"(F)F)F)F6/F.;\"\"!,$*&\"\"#F6%#PiGF)F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"%X@\"\"\"\"'s58!\"\"%#PiGF&F&#F &\"\"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "H:=MF([x,y,z,w] ,[x^2,x*w,w,z^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&f*6&%\"x G%\"yG%\"zG%\"wG6\"6$%)operatorG%&arrowGF,*$)9$\"\"#\"\"\"F,F,F,f*F'F, F-F,*&F2F49'F4F,F,F,f*F'F,F-F,F7F,F,F,f*F'F,F-F,*$)9&F3F4F,F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r:=MF(t,[sin(t),cos(t),sin(t ),cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7&%$sinG%$cosGF&F '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "liv(H,r,t=0..Pi/2,'ste p'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$)-%$cosG6#%\"tG\" \"#\"\"\"F.-%$sinGF+!\"\"*&F/F.F(F.F./F,;\"\"!,$*&F-F1%#PiGF.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%%EvalG6$,**&#\"\"\"\"\"#F+*&-% $cosG6#%\"tGF+-%$sinGF0F+F+F+*&F,!\"\"F1F+F+F.F+*&#F+\"\"$F+*$)F.F8F+F +F5/F1;\"\"!,$*&F,F5%#PiGF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~ G,&*&\"\"%!\"\"%#PiG\"\"\"F*#\"\"#\"\"$F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "Ve cCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MakeFunction" 2 "MakeFunctio n" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multipleint" 2 "Mu ltipleint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LineIntScalar" 2 "LineIn tScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntScalar" 2 "Surfa ceIntScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntVector" 2 "S urfaceIntVector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VectorCalculus[Lin eInt]" 2 "VectorCalculus[LineInt]" "" }{TEXT -1 1 "." }}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } RA 11 "" 1 "" {XPPMATH 20 "6#/%\"~ G,&*&\"\"%!\"\"%#PiG\"\"\"F*#\"\"#\"\"$F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2003 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "VecCalc" 2 "Ve cCalc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "MakeFunction" 2 "MakeFunctio n" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multipleint" 2 "Mu ltipleint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LineIntScalar" 2 "LineIn tScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntScalar" 2 "Surfa ceIntScalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "SurfaceIntVector" 2 "S urfaceIntVector" "" }{TEXT -1 2with""""""""""""""""""""""""""""""""""""withou"without"""wl"wo"won"work'"""""""""written"xampl ""xecut ""xgW\"""""""" """""""""""""xgf """"xn"yasskin)" """""""""""""""""""""""""""""""ygWN""""""""" """"""""""""ygf """""yl"you7"""""""""""""your"yp"ype ""yse"yxp"za"zdhd"zero"zgO>"""""""""""""""""""zgf """"""gcolumngcossincosgsingffdfpipigfhgwgsingcosgfsteevalgtgfpigcopyrightarthurbelmontphilipasskindepartmmathematictexauniversitalsoveccalcmakefunctiointmultipleintmultipleintlineintscalarlineintscalarsurfaceintscalarsurfaceintscalarsurfaceintvectorurfaceintvectorvccommandlivlineintvectorinertcallsequencneintvectorinertvarrnglislineintvectorinertparameterformlistvariablarrownotatcurvdefinedparametintegratrangoveroptionalindicatdisplaydescriptfirstargumsecondthirdevaluatusingvaluevalfcalculatwithouthincludhilealllineintvectorinertreturninertmultiplwhileyoudoneedquotaroundassignsimilarommandvectorcalcululineintvectorcalculupackagwithdomainchosenpathhowevusesexprsioninsteadcannotnorshowrequirspecificatcoordinatsystemthespartcanameonlyperformithfunctalwayaccesslongformsexamplmakefunctfgxgygzgoperatorgarrowgfufrgtgperatorgintgudfmfggrtablegmatrixgarrowgfvector!LeadingPrincipalMinorDeterminantsGMathematics/Packages/VecCalc/Commands/LeadingPrincipalMinorDeterminants!LeadingPrincipalMinorDeterminants)Mathematics/Packages/VecCalc/Aliases/LPMDLength#Mathematics/Packages/VecCalc/LengthLength,Mathematics/Packages/VecCalc/Commands/Length LineIntScalar*Mathematics/Packages/VecCalc/LineIntScalar LineIntScalar3Mathematics/Packages/VecCalc/Commands/LineIntScalar LineIntScalar8Mathematics/Packages/VecCalc/Commands/LineIntScalarInert LineIntScalar(Mathematics/Packages/VecCalc/Aliases/Lis LineIntScalar(Mathematics/Packages/VecCalc/Aliases/lis LineIntVector*Mathematics/Packages/VecCalc/LineIntVector LineIntVector3Mathematics/Packages/VecCalc/Commands/LineIntVector LineIntVector8Mathematics/Packages/VecCalc/Commands/LineIntVectorInert LineIntVector(Mathematics/Packages/VecCalc/Aliases/Liv LineIntVector(Mathematics/Packages/VecCalc/Aliases/liv MakeFunction)Mathematics/Packages/VecCalc/MakeFunction MakeFunction2Mathematics/Packages/VecCalc/Commands/MakeFunction MakeFunction)Mathematics/Packages/VecCalc/Commands/&->} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixe d Width" -1 256 1 {CSTYLE "" -1 -1 "Courier"