- ),he dot product of the two vectors u and v." }}{PARA 15 "" 0 "" {TEXT -1 100 "vec_calc[dot] is a modification of linalg[dotprod] \+ where the 'orthogonal' option is always selected." }}{PARA 15 "" 0 "" {TEXT -1 245 "The function dot and the operator &. are part of the vec _calc package, and so can be used by name only after performing the co mmand with(vec_calc) or with(vec_calc, dot, `&.`). The function can a lways be accessed in the long form vec_calc[dot]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u:=[1,2,3]; v:=[a,b,c];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\"\"#%\"cG\" \"$" }}}{EXC<_R{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "MS Sans Serif" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bulle t Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Dash Item" 0 16 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 16 3 } {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 }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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 288 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 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 21 " \+ Introduction to the " }{TEXT 256 8 "vec_calc" }{TEXT -1 23 " Package - - Version 4.3" }}{PARA 0 "" 0 "" {TEXT -1 49 "The possible HELP pages \+ are listed at the bottom." }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequ ences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 17 " command(ar gs) " }}{PARA 258 "" 0 "" {TEXT -1 26 " vec_calc[command](args)" }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 125 "The vec_calc package is a collection of commands designed to simplify calculations which arise from vector ca lculus problems." }}{PARA 15 "" 0 "" {TEXT -1 49 "To load the package, be sure the system variable " }{HYPERLNK 17 "libname" 2 "libname" "" }{TEXT -1 101 " includes the path to the directory containing the pack age. Then execute the command with(vec_calc);" }}{PARA 15 "" 0 "" {TEXT -1 54 "The vec_calc package automatically loads the packages:" } }{PARA 17 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "student" 2 "student" " " }{TEXT -1 3 " " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 3 " " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 54 "The vec_calc package automatically loads the command s:" }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "mtaylor" 2 "mtayl or" "" }{TEXT -1 3 " " }{HYPERLNK 17 "polar" 2 "polar" "" }{TEXT -1 3 " " }{HYPERLNK 17 "unassign" 2 "unassign" "" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 111 "Many of the vec_calc commands have shor ter aliases which become available after executing the vec_calc comman d " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 159 "The vec_calc commands are divided into \+ several groups. These commands are listed below by group and are foll owed by the alias in parentheses, if there is one." }}{PARA 16 "" 0 " " {TEXT -1 88 "The commands to perform function definition, vector man ipulation and simplification are:" }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "MF" 2 "makefunction" "" }{TEXT -1 9 " ) " } {HYPERLNK 17 "evall" 2 "evall" "" }{TEXT -1 11 " " } {HYPERLNK 17 "ss" 2 "ss" "" }{TEXT -1 2 " " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "dot" 2 "dot" "" }{TEXT -1 4 " or " } {HYPERLNK 17 "&." 2 "dot" "" }{TEXT -1 18 " " } {HYPERLNK 17 "cross" 2 "cross" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "&x " 2 "cross" "" }{TEXT -1 5 " " }{HYPERLNK 17 "len" 2 "len" "" } {TEXT -1 1 " " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "vc_al iases" 2 "vc_aliases" "" }{TEXT -1 37 " \+ " }}{PARA 16 "" 0 "" {TEXT -1 39 "The commands to change coordin ates are:" }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "deg2rad" 2 "deg2rad" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "d2r" 2 "deg2rad" "" } {TEXT -1 8 " ) " }{HYPERLNK 17 "rad2deg" 2 "deg2rad" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2d" 2 "deg2rad" "" }{TEXT -1 6 " ) " }} {PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "polar2rect" 2 "CoordCo nversion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "p2r" 2 "CoordConversi on2D" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "rect2polar" 2 "CoordConver sion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2p" 2 "CoordConversion2D " "" }{TEXT -1 3 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{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 5 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{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 5 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "sph2cyl " 2 "CoordConversion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "s2c" 2 "C oordConversion3D" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "cyl2sph" 2 "CoordConversion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "c2s" 2 "Coord Conversion3D" "" }{TEXT -1 6 " ) " }}{PARA 16 "" 0 "" {TEXT -1 36 " The commands to analyse a curve are:" }}{PARA 17 "" 0 "" {TEXT -1 2 " \+ " }{HYPERLNK 17 "curve_velocity" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Cv" 2 "Curve" "" }{TEXT -1 6 " ) " }{HYPERLNK 17 "cur ve_acceleration" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Ca" 2 "Curve" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "curve_jerk" 2 "Curve" " " }{TEXT -1 4 " ( " }{HYPERLNK 17 "Cj" 2 "Curve" "" }{TEXT -1 7 " ) \+ " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "curve_tangent" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CT" 2 "Curve" "" } {TEXT -1 7 " ) " }{HYPERLNK 17 "curve_normal" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CN" 2 "Curve" "" }{TEXT -1 11 " ) \+ " }{HYPERLNK 17 "curve_binormal" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPE RLNK 17 "CB" 2 "Curve" "" }{TEXT -1 3 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "curve_arclength" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CL" 2 "Curve" "" }{TEXT -1 5 " ) " } {HYPERLNK 17 "curve_curvature" 2 "Curve" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Ck" 2 "Curve" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "c urve_torsion" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Ct" 2 "Cu rve" "" }{TEXT -1 4 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " } {HYPERLNK 17 "curve_tangential_acceleration" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CaT" 2 "Curve" "" }{TEXT -1 6 " ) " } {HYPERLNK 17 "curve_normal_acceleration" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CaN" 2 "Curve" "" }{TEXT -1 4 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "curve_forget" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Cforget" 2 "Curve" "" }{TEXT -1 55 " ) \+ " }}{PARA 16 "" 0 "" {TEXT -1 45 "The commands for differential operations are:" } }{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 6 " " }{HYPERLNK 17 "DIV" 2 "DIV" "" }{TEXT -1 12 " " }{HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 10 " " }{HYPERLNK 17 "LAP" 2 "LAP" "" }{TEXT -1 15 " " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "HESS" 2 "HESS" "" }{TEXT -1 6 " " }{HYPERLNK 17 "leading_principal_minor_determinants" 2 "leading_princi pal_minor_determinants" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "LPMD" 2 " leading_principal_minor_determinants" "" }{TEXT -1 3 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "JAC" 2 "JAC" "" }{TEXT -1 7 " \+ " }{HYPERLNK 17 "JAC_DET" 2 "JAC_DET" "" }{TEXT -1 8 " " } {HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 11 " " }{HYPERLNK 17 "VEC_POT" 2 "VEC_POT" "" }{TEXT -1 11 " " }}{PARA 16 "" 0 "" {TEXT -1 41 "The commands for integral operations are:" }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Multipleint" 2 "Multipleint " "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Muint" 2 "Multipleint" "" } {TEXT -1 10 " ) " }{HYPERLNK 17 "multipleint" 2 "Multipleint" " " }{TEXT -1 4 " ( " }{HYPERLNK 17 "muint" 2 "Multipleint" "" }{TEXT -1 8 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Line _int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 " Lis" 2 "Line_int_scalar" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "line _int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 " lis" 2 "Line_int_scalar" "" }{TEXT -1 6 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" " " }{TEXT -1 4 " ( " }{HYPERLNK 17 "Liv" 2 "Line_int_vector" "" } {TEXT -1 8 " ) " }{HYPERLNK 17 "line_int_vector" 2 "Line_int_vect or" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "liv" 2 "Line_int_vector" "" } {TEXT -1 6 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 " Surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 4 " ( " } {HYPERLNK 17 "Sis" 2 "Surface_int_scalar" "" }{TEXT -1 5 " ) " } {HYPERLNK 17 "surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "sis" 2 "Surface_int_scalar" "" }{TEXT -1 3 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Surface_int_vecto r" 2 "Surface_int_vector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Siv" 2 "Surface_int_vector" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "surface_int _vector" 2 "Surface_int_vector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "s iv" 2 "Surface_int_vector" "" }{TEXT -1 3 " ) " }}{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 pro duct of two vectors A and B, use " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A:=[x,y,z];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "B:=[1,2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(A,B );" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 17 "Acknowledgements:" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 165 "The commands in this p ackage are used extensively throughout the text \"Vector Calculus with Maple V\" by Arthur Belmonte and Philip B. Yasskin, to be published, \+ 1997." }}{PARA 15 "" 0 "" {TEXT -1 289 "The vec_calc commands were wri tten by A. Belmonte and P. Yasskin. The commands were organized into \+ a package by James Warren and P. Yasskin. The help pages were first w ritten by David Arnold, J. Warren and P. Yasskin and converted to Rele ase 4 by Ken Parker, Jared Teslow and P. Yasskin." }}{PARA 15 "" 0 "" {TEXT -1 135 "@ Copyright 1995-97 by Arthur Belmonte and Philip B. Yas skin, Department of Mathematics, 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 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_Min" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" }{TEXT -1 39 ". You are supposed to be able to type ?" }{TEXT 259 7 "c ommand" }{TEXT -1 14 " or ?vec_calc[" }{TEXT 257 7 "command" }{TEXT -1 9 "] (where " }{TEXT 258 7 "command" }{TEXT -1 168 " is from the ab ove list), but this does not work for all command names in Release 4. \+ However, help is available on all commands. See the Help Cross-Refer encing below>" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Help Cross-Refe rencing:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 250 "At present I am having trouble getting the help system to properly cross reference the help pages. So here is the list of all help pages and the list o f help topics which should point to each of them. Just click on one o f the main help pages below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 21 "vec_calc -- this page" }}{PARA 15 "" 0 "" {HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "makefunction" 2 "makefunction" "" }}{PARA 293 "" 0 "" {TEXT -1 5 " MF" }}{PARA 15 "" 0 "" {HYPERLNK 17 "evall" 2 "evall" " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "ss" 2 "ss" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "dot" 2 "dot" "" }}{PARA 292 "" 0 "" {TEXT -1 5 " &." } }{PARA 15 "" 0 "" {HYPERLNK 17 "cross" 2 "cross" "" }}{PARA 291 "" 0 " " {TEXT -1 5 " &x" }}{PARA 15 "" 0 "" {HYPERLNK 17 "len" 2 "len" "" }}{PARA 0 "" 0 "" {TEXT -1 70 "_______________________________________ _______________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "deg2 rad" 2 "deg2rad" "" }}{PARA 289 "" 0 "" {TEXT -1 16 " d2r " }}{PARA 290 "" 0 "" {TEXT -1 16 " rad2deg r2d" }}{PARA 15 "" 0 " " {HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }}{PARA 287 "" 0 "" {TEXT -1 19 " polar2rect p2r" }}{PARA 288 "" 0 "" {TEXT -1 19 " rect2polar r2p" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Coo rdConversion3D" 2 "CoordConversion3D" "" }}{PARA 281 "" 0 "" {TEXT -1 17 " cyl2rect c2r" }}{PARA 282 "" 0 "" {TEXT -1 17 " rect2cyl \+ r2c" }}{PARA 283 "" 0 "" {TEXT -1 17 " sph2rect s2r" }}{PARA 284 " " 0 "" {TEXT -1 17 " rect2sph r2s" }}{PARA 285 "" 0 "" {TEXT -1 17 " sph2cyl s2c" }}{PARA 286 "" 0 "" {TEXT -1 17 " cyl2sph \+ c2s" }}{PARA 0 "" 0 "" {TEXT -1 70 "__________________________________ ____________________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Curve" 2 "Curve" "" }}{PARA 270 "" 0 "" {TEXT -1 9 " frenet" }} {PARA 294 "" 0 "" {TEXT -1 36 " curve_velocity Cv" } }{PARA 271 "" 0 "" {TEXT -1 36 " curve_acceleration Ca" }}{PARA 272 "" 0 "" {TEXT -1 36 " curve_jerk Cj " }}{PARA 273 "" 0 "" {TEXT -1 36 " curve_tangent C T" }}{PARA 274 "" 0 "" {TEXT -1 36 " curve_normal \+ CN" }}{PARA 275 "" 0 "" {TEXT -1 36 " curve_binormal \+ CB" }}{PARA 276 "" 0 "" {TEXT -1 36 " curve_arclength \+ CL" }}{PARA 277 "" 0 "" {TEXT -1 36 " curve_curvature \+ Ck" }}{PARA 278 "" 0 "" {TEXT -1 36 " curve_torsion \+ Ct" }}{PARA 279 "" 0 "" {TEXT -1 37 " curve_tangential_accelerat ion CaT" }}{PARA 280 "" 0 "" {TEXT -1 37 " curve_normal_acceleratio n CaN" }}{PARA 15 "" 0 "" {HYPERLNK 17 "curve_forget" 2 "curve_fo rget" "" }}{PARA 269 "" 0 "" {TEXT -1 10 " Cforget" }}{PARA 0 "" 0 " " {TEXT -1 70 "_______________________________________________________ _______________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Diffops" 2 "Diffops " "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_Min " "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "GRAD" 2 "GRAD" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "DIV" 2 "DIV" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "CU RL" 2 "CURL" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "LAP" 2 "LAP" "" }} {PARA 15 "" 0 "" {HYPERLNK 17 "HESS" 2 "HESS" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "leading_principal_minor_determinants" 2 "leading_princip al_minor_determinants" "" }}{PARA 268 "" 0 "" {TEXT -1 7 " LPMD" }} {PARA 15 "" 0 "" {HYPERLNK 17 "JAC" 2 "JAC" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "JAC_DET" 2 "JAC_DET" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "POT" 2 "POT" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "VEC_POT" 2 "VEC_POT " "" }}{PARA 0 "" 0 "" {TEXT -1 70 "__________________________________ ____________________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Multipleint" 2 "Multipleint" "" }}{PARA 259 "" 0 "" {TEXT -1 22 " \+ Muint" }}{PARA 259 "" 0 "" {TEXT -1 22 " multipleint \+ muint" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Line_int_scalar" 2 "Line_int_s calar" "" }}{PARA 260 "" 0 "" {TEXT -1 24 " Lis" } }{PARA 261 "" 0 "" {TEXT -1 24 " line_int_scalar lis" }}{PARA 15 " " 0 "" {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }}{PARA 262 "" 0 "" {TEXT -1 24 " Liv" }}{PARA 263 "" 0 " " {TEXT -1 24 " line_int_vector liv" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }}{PARA 264 "" 0 "" {TEXT -1 27 "  Sis" }}{PARA 265 "" 0 "" {TEXT -1 27 " surface_int_scalar sis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "S urface_int_vector" 2 "Surface_int_vector" "" }}{PARA 266 "" 0 "" {TEXT -1 27 " Siv" }}{PARA 267 "" 0 "" {TEXT -1 27 " surface_int_vector siv" }}{PARA 0 "" 0 "" {TEXT -1 70 "___ ___________________________________________________________________" } }}} X{TEXT `TEXTh{HYPERLNK pHYPERLNKx{TEXT lis" }}{PARA 15 " " 0 "" {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }}{PARA 262 "" 0 "" {TEXT -1 24 " Liv" }}{PARA 263 "" 0 " " {TEXT -1 24 " line_int_vector liv" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }}{PARA 264 "" 0 "" {TEXT -1 27 " <vec_calc< vc_aliases< makefunction<evall<ss<dot<cross<len<deg2rad<CoordConversion2D<CoordConversion3D<Curve< curve_forget<Diffops< Multi_Max_Min<GRAD<DIV<CURL<LAP<HESS<$leading_principal_minor_determinants<JAC<JAC_DET<POT<VEC_POT< Multipleint=Line_int_scalar=Line_int_vector=Surface_int_scalar=Surface_int_vector<commandveccalcvcaliassetssomepackagcallsequencdescriptdefindoneusingmaplaliaoutputincludallprevioumfmakefunctcvcurvvelocitcttangcaacceleratcnnormalcjjerkcbbinormalckcurvaturcattangentialtorscanclarclengthcforgetforgetdegradpolarrectcylsphlpmdleadprincipalminordeterminantmuintmultipleintlislineintscalarlivvectorsissurfacsivpartusedformonlyafterperformwithexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalso< extensive<first<follow<forget <<frenet<funct<gett<grad<group<have<having<help < here<hess<howev<includ <<int <<integral<into< introduct<jac<jame<jared<jerk < " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6F%\"IG%&PointG%#MFG%#CvG%#CaG%#CjG%#CTG%#CNG%#CBG%#CkG%# CtG%#CLG%$CaTG%$CaNG%(CforgetG%$d2rG%$r2dG%$p2rG%$r2pG%$c2rG%$r2cG%$s2 rG%$r2sG%$s2cG%$c2sG%&MuintG%&muintG%%LPMDG%$LisG%$lis'G%$LivG%$livG%$S isG%$sisG%$SivG%$sivG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f: =MF([x,y,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(* $9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8F3F+F +:F'F+F,F+*&F1F2F8F6F+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%:6%%\"xG%\"yG%\"zG6\"6$%)opera torG%&arrowGF+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+ F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyrig ht 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department \+ of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 8 "S ee Also" }{TEXT -1 2 ": " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 2 ". " }} }} (F([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%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(* $9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8F3F+F +:F'F+F,F+*&F1F2F8F6F+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%:6%%\"xG%\"yG%\"zG6\"6$%)opera torG%&arrowGF+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+ F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyrig ht 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department \+ of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 8 "S ee Also" }{TEXT -1 2 ": " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 2 ". " }} }} (<<p @ ԺequationJN D Ժ= " 0 "" {MPLTEXT 1 0 27 "with(vec_ca lc): vc_aliases:" }}}{PARA 0 "" 0 "" {TEXT -1 34 "A scalar function of one variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=makefun ction(t,t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%) operatorG%&arrowGF(*$9$\"\"#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 49 "A ve ctor function of one variable (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%:6#%\"tG6\"6$%)operatorG%&arrowGF)9$F)F):F'F)F* F)*$F-\"\"#F)F):F'F)F*F)*$F-\"\"$F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 56 "A scalar function of several variables (e.g. a density):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f:=makefunction([x,y,z,t]1,x^2*sin(t *y)+ln(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6&%\"xG%\"yG%\"z G%\"tG6\"6$%)operatorG%&arrowGF+,&*&9$\"\"#-%$sinG6#*&9'\"\"\"9%F8F8F8 -%#lnG6#9&F8F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 116 "A vector function o f several variables (e.g. a vector field or a coordinate transformatio n or a parametric surface):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f:=makefunction([u,v],[(u+v)/2, (u-v)/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$:6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF*,&9$#\" \"\"\"\"#9%F0F*F*:F'F*F+F*,&F/F0F3#!\"\"F2F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 46 "A list of lists function of several variables:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f:=MF([x,y,t],[[x^2,x+t,y-x^3*t^2], [t,y,x^2]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG7$7%:6%%\"xG%\"y G%\"tG6\"6$%)operatorG%&arrowGF,*$9$\"\"#F,F,:F(F,F-F,,&F1\"\"\"9&F5F, F,:F(F,F-F,,&9%F5*&F1\"\"$F6F2!\"\"F,F,7%:F(F,F-F,F6F,F,:F(F,F-F,F9F,F ,:F(F,F-F,F0F,F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belm2onte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ". " }}}} +v)/2, (u-v)/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$:6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF*,&9$#\" \"\"\"\"#9%F0F*F*:F'F*F+F*,&F/F0F3#!\"\"F2F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 46 "A list of lists function of several variables:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f:=MF([x,y,t],[[x^2,x+t,y-x^3*t^2], [t,y,x^2]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG7$7%:6%%\"xG%\"y G%\"tG6\"6$%)operatorG%&arrowGF,*$9$\"\"#F,F,:F(F,F-F,,&F1\"\"\"9&F5F, F,:F(F,F-F,,&9%F5*&F1\"\"$F6F2!\"\"F,F,7%:F(F,F-F,F6F,F,:F(F,F-F,F9F,F ,:F(F,F-F,F0F,F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belm2<helpdifferentialoperstorusingveccalcpackagfunctionmakefunctmakearrowdefinfunctgradcalculatgradidivdivergenccurllaplaplacianhesshessianleadprincipalminordeterminantjacjacobianmatrixdetpotfindscalarpotentialexistvectoraliasthescanusedafterexecutvccommandmflpmdcallsequencoutfcnvarsparameternamelistrepresentindependvariablexpressvalufieldformmustsquarcoordinattransformatequalreturnoptionaldescriptdesignwithdefinitdifferentiatantiincludparametrsurfaceachownpageexampluseyoufirstcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvmultimaxminity " }}{PARA 0 "" 0 "" {TEXT 26 9 "S ee Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 1 "." }}}} 6 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0<functveccalcleadprincipalminordeterminantcalculatmatrixaliacanusedafterexecutvcaliascommandlpmdcallsequencparametersquarlistexpressdescriptsubmatrictopleftcorncomputdisplayreturntheslinearalgebrapositdefinitallmultivariablcalculucriticalpointlocalminimumhessianatmaximumalternatsignbeginnwithnegatpartpackagformonlyperformalwayaccesslongexampldmsubmatrixdetcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsolinalgmultimaxminhess. RETURNJC Re SearchTextJ:5  absJFH addJFtGT addressofJS aliasJO sm~ׇh anamesJl$ appendtoJju4 arrayJHfhh < {VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 18 "vec_calc[evall] - " }{TEXT6 -1 44 "Evaluate a List byP erforming Vector Algebra " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequ ences:" }{TEXT -1 1 "\n" }{TEXT 256 40 " evall(expr) vec_calc[ev all](expr)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 " \n" }{TEXT 257 12 " expr - " }{TEXT -1 51 "an algebraic expression involving lists or vectors." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 " Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 69 "evall(ex pr) is entirely equivalent to convert(evalm(expr),listlist). " }} {PARA 15 "" 0 "" {TEXT -1 199 "evall performs calculations on lists an d lists of lists by converting the expression into vectors and matrice s, simplifying the expression, and converting the result back to a lis t or list of lists. " }}{PARA 15 "" 0 "" {TEXT -1 229 "This function i s part of the vec_calc package, and so can be used in the form evall o nly after performing the command with(vec_calc) or with(vec_calc, eval l). The function can always be accessed in the long form vec_calc[eva ll]7." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}} {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%,$%\"xGF),$%\"yGF),$%\"zGF)F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evall(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%,&*$\"\"##\"\"\"F&F(%\"xG\"\"$,&F%F&%\"yGF*,&F %F*%\"zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalm" 2 "evalm" "" }{TEXT -1 2 ", " 8}{HYPERLNK 17 "convert" 2 "co nvert" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "convert[listlist]" 2 "conver t[listlist]" "" }{TEXT -1 2 ". " }}}} equationJN <) 3]; v:=[x,y,z];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{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%\"\"$F'F'7%,$%\"xGF),$%\"yGF),$%\"zGF)F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evall(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%,&*$\"\"##\"\"\"F&F(%\"xG\"\"$,&F%F&%\"yGF*,&F %F*%\"zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalm" 2 "evalm" "" }{TEXT -1 2 ", " 8 " 0 "" {MPLTEXT 1 0 15 "with(vec_ca lc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g:=(x^3-x^6)^(1/3); simplify(g); ss(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*$,&*$% \"xG\"\"$\"\"\"*$F(\"\"'!\"\"#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$,$*&%\"xG\"\"$,&!\"\"\"\"\"*$F&F'F*F*F)#F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&F%F%*$F$\"\"$!\"\"#F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=sqrt(x^2); simplify(f); ss(f); # incorrect if x is negative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$ *$%\"xG\"\"##\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#% \"xG\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 "<, " }{HYPERLNK 17 "simplify [radical]" 2 "simplify[radical]" "" }{TEXT -1 2 ". " }}}}  ,getuserinterface6)^(1/3); simplify(g); ss(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*$,&*$% \"xG\"\"$\"\"\"*$F(\"\"'!\"\"#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$,$*&%\"xG\"\"$,&!\"\"\"\"\"*$F&F'F*F*F)#F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&F%F%*$F$\"\"$!\"\"#F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=sqrt(x^2); simplify(f); ss(f); # incorrect if x is negative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$ *$%\"xG\"\"##\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#% \"xG\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 "<<commandveccalcmakefunctmakefunctusingpackagaliacanusedafterexecutvcaliasmfcallsequencoutparameternamelistrepresentindependvariablexpressarrascalarvectorvalunestarraydescriptdefinmorearrowformproducfunctionconsistwithsamestructurpartonlyperformalwayaccesslongexamplcurvseveraldensitsinlnfieldcoordinattransformatparametricsurfaccopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffop " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u:=[1,2,3]; v:=[a,b,c];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\"\"#%\"cG\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &. v;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\"\"#%\"cG\"\"$" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "dotprod" 2 "dotprod" "" }{TEXT -1 2 ", " }A{HYPERLNK 17 "cross" 2 " cross" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "len" 2 "len" "" }{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 "." }} }} %\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\"\"#%\"cG\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &. v;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\"\"#%\"cG\"\"$" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "dotprod" 2 "dotprod" "" }{TEXT -1 2 ", " }A<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1C "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "B ullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 18 "vec_calc[cross] - " }{TEXT -1 45 "Calculates the Cros s Product of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 10 "Operator: \+ " }{TEXT 259 18 "vec_calc[&x] - " }{TEXT -1 44 "Calculates the Cros s Product of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequ ences:" }{TEXT -1 1 "\n" }{TEXT 256 51 " cross(u, v) u &x v \+ vec_calc[cross](u, v)" }}{PARA 7 "" 0 "" {TEXT -1 142 "CAUTION: There \+ must be a space after the letter x in &x, or else Maple will think the subsequent letters are part of the name of the operator." }}{PARA 0 " " 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 257 12 " u , v - " }{TEXT -1 43 "lists or vectors, each with three elements." } }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "DescriptioDn:" }{TEXT -1 1 " " } }{PARA 15 "" 0 "" {TEXT -1 113 "cross(u,v) and u &x v calculate the cr oss product of two 3-dimensional vectors and returns the answer as a l ist. " }}{PARA 15 "" 0 "" {TEXT -1 93 "vec_calc[cross] is a modificati on of linalg[crossprod] to return a list instead of a vector. " }} {PARA 15 "" 0 "" {TEXT -1 251 "The function cross and the operator &x \+ are part of the vec_calc package, and so can be used by name only afte r performing the command with(vec_calc) or with(vec_calc, cross, `&x`) . The function can always be accessed in the long form vec_calc[cross ]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{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%\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "cross(u,v); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%!\" $\"\"'F$"E }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &x v;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%!\"$\"\"'F$" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 " vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "crossprod" 2 "crossprod" "" }{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 "operators[precedence]" "" } {TEXT -1 1 "." }}}} TEXT 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%\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "cross(u,v); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%!\" $\"\"'F$"E<functveccalcsssymbolicsimplificatexpresscallsequencexprparameteranydescriptevallentireequivalsimplifparametwelldocumentbutbasicalpermitrealansweravoidcomplexunfortunatemayalsolosesomecautadvispartpackagcanusedformonlyafterperformcommandwithalwayaccesslongexamplsqrtincorrectnegatcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitradical<`functveccalcdotcomputproductvectoroperatorcallsequencparameterlistsamelengthdescriptcalculatmodificatlinalgdotprodorthogonaloptionalwayselectpartpackagcanusednameonlyafterperformcommandwithaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocroslenneutralprecedenc" {TEXT -1 10 " CL = " }{HYPERLNK 17 "curve_arclength" 2 "Curve" "" }{TEXT -1 16 " Cforget = " } {HYPERLNK 17 "curve_forget" 2 "curve_forget" "" }{TEXT -1 18 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " d2r = "  " 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 23 "with(vec_calc): len(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(* $%\"aG\"\"#\"\"\"*$%\"bGF'F(*$%\"cGF'F(#F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(linalg): norm(v,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(*$-%$absG6#%\"aG\"\"#\"\"\"*$-F'6#%\"bGF*F+*$-F'6#% \"cGF*F+#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:=[sin(t) ,cos(t)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7$-%$sinG6#%\"tG-%$ cosGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "len(u); simplify( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$-%$sinG6#%\"tG\"\"#\"\" \"*$-%$cosGF(F*F+#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT J1 0 42 "norm(u,2); simplify(\"); # t may be complex" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$-%$absG6#-%$s inG6#%\"tG\"\"#\"\"\"*$-F'6#-%$cosGF+F-F.#F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$-%$absG6#-%$sinG6#%\"tG\"\"#\"\"\"*$-F'6#-%$cosGF +F-F.#F.F-" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995- 97 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathe matics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also: " }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[norm]" 2 "linalg[norm]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "dot" 2 "dot" "" }{TEXT -1 1 "." }}}} )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7$-%$sinG6#%\"tG-%$ cosGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "len(u); simplify( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&*$-%$sinG6#%\"tG\"\"#\"\" \"*$-%$cosGF(F*F+#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT Jexecut?<<<<<<<<<<<====exist<<<exp<<<<expr <<express?<<<<<<<<<<<<<<< extensive<extremiz<fail<fals <<fcn<<<<==field+-<<<<<<====fifth<final<find <<finish<first3<<<<<<<<====float<<<follow <<forget<<<< forms?<<<<<<<<<<<<<<<<<<<<<<<<====formula << H int/seriesJ< 6V@  H iolibJJObt  H iquoJ+y  H iremJQʅ   teslow<test<texa{<<<<<<<<<<<<<<<<<<<<<<<<<<====text<th<thefunct<their<==them <<thes?<<<<<<<<<<<====theta8<<<<<==they <<think<third<==== throughout<time<top<topic<tors<<< transformat <<<< tripleint<troubl<\"xG%\"y G%\"tG6\"6$%)operatorG%&arrowGF,*$9$\"\"#F,F,:F(F,F-F,,&F1\"\"\"9&F5F, F,:F(F,F-F,,&9%F5*&F1\"\"$F6F2!\"\"F,F,7%:F(F,F-F,F6F,F,:F(F,F-F,F9F,F ,:F(F,F-F,F0F,F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belm2 <{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PST7 0 } {PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 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 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 23 " vec_calc[deg2rad] - " }{TEXT -1 40 "Converts An gles from Degrees to Radians " }}{PARA 0 "" 0 "" {TEXT 257 23 " vec_ calc[rad2deg] - " }{TEXT -1 39 "Converts Angles from Radians to Degree s" }}{PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliase s can be used after execution of the " }{HYPERLNK 17 "vc_aliases" 2 "v c_aliases" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 18 " d2r = deg2rad" }}{PARA 259 "" 0 "" {TEXT -1 18 " r2d = rad2d eg" }}{POARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 61 " deg2rad(theta) d2r(theta) \+ vec_calc[deg2rad](theta)" }}{PARA 256 "" 0 "" {TEXT -1 61 " rad2de g(theta) r2d(theta) vec_calc[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 a ngle " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 75 "deg2rad converts angles measu red in degrees to angles measured in radians. " }}{PARA 15 "" 0 "" {TEXT -1 75 "rad2deg converts angles measured in radians to angles mea sured in degrees. " }}{PARA 15 "" 0 "" {TEXT -1 166 "If theta contains any floating point decimal numbers, then deg2rad and rad2deg return d ecimal answers. Otherwise, they return exact numbers or symbolic expr essions. " }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part o f the vec_calc package, and so canP be used by name only after performi ng the command with(vec_calc) or with(vec_calc,function). The functio ns can always be accessed in the long forms vec_calc[function]. The a liases can be used only after performing the command " }{HYPERLNK 17 " vc_aliases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "deg2rad(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&% \"aG\"\"\"%#PiGF&#F&\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "rad2deg(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG\"\"\"%#Pi G!\"\"\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2rad(45); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{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 "deg2 rad(45.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N;)R&y!#5" }}}{EXCHG {PAQRA 0 "> " 0 "" {MPLTEXT 1 0 13 "rad2deg(1.); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]zdHd!\")" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n D epartment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D " "" }{TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConver sion3D" "" }{TEXT 26 1 "." }}}} @˸ .'"XT 1 0 11 "rad2deg(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG\"\"\"%#Pi G!\"\"\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2rad(45); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rad2deg(Pi/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "deg2 rad(45.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N;)R&y!#5" }}}{EXCHG {PAQ=8functionveccalclineintscalardisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandliscallsequencfcnvarrngparameterfunctvariablarrownotatcurvformlistdefinparametintegratrangoveroptionalindicatdescriptfirstargumsecondthirdevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignthespartpackagnameonlyperformalwayaccesslongexamplmakefunctsincospimflncopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintvectorsurfac       0    l p ` D     X           = 0" "1\"\"!%\"rG" }{TEXT -1 5 " a nd " }{XPPEDIT 19 1 " -Pi < theta " "2,$%#PiG!\"\"%&thetaG" }{XPPEDIT 19 1 "``< Pi" "2%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 97 "Maple's aYrctan function with 2 arguments is designed to produce exactly what is needed for theta." }}{PARA 15 "" 0 "" {TEXT -1 89 "Th ese functions return floating point decimal numbers if the input conta ins any decimals." }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions ar e part of the vec_calc package, and so can be used by name only after \+ performing the command with(vec_calc) or with(vec_calc,function). The functions can always be accessed in the long forms vec_calc[function] . The aliases can be used only after performing the command " } {HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "polar2rect([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"aG\"\"\"-%$cosG6#%\"bGF&*&F%F&-%$sinGF)F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2polar([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$,&*$%\"aG\"\"#\"\"\"*$%\"bZGF(F)#F)F(-%'a rctanG6$F+F'" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 199 5-97 by Arthur Belmonte and Philip B. Yasskin\n Department of Mat hematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Al so: " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT 26 2 ", " } {HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[deg2rad]" 2 "vec_calc[deg2rad]" "" } {TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[rad2deg]" 2 "vec_calc[rad2deg] " "" }{TEXT -1 2 ". " }}}} A 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "polar2rect([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"aG\"\"\"-%$cosG6#%\"bGF&*&F%F&-%$sinGF)F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2polar([a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$,&*$%\"aG\"\"#\"\"\"*$%\"bZ copyright{<<<<<<<<<<<<<<<<<<<<<<<<<<====corn< correspond<cos/<<<<<<<====counterclockwis <<critical <<critpt<cros<<< crossprod<ct <<<curl<<<<<<<curv+h<< <= 0;" "1 \"\"!%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 19 1 " -Pi < theta;" "2,$%#P iG!\"\"%&thetaG" }{XPPEDIT 19 1 "``<= Pi;" "1%!G%#PiG" }{TEXT -1 4 ". \+ " }}{PARA 15 "" 0 "" {TEXT -1 97 "Maple's arctan function with 2 arg uments is designed to produce exactly what is needed for theta." }} {PARA 15 "" 0 "" {TEXT -1 104 "sph2recht converts spherical coordinates to rectangular coordinates using the formulas:\n " } {XPPEDIT 19 1 "x = rho * sin(phi)*cos(theta);" "/%\"xG*(%$rhoG\"\"\"-% $sinG6#%$phiGF&-%$cosG6#%&thetaGF&" }{TEXT -1 12 " " } {XPPEDIT 19 1 "y = rho *sin(phi) *sin(theta) ;" "/%\"yG*(%$rhoG\"\"\"- %$sinG6#%$phiGF&-F(6#%&thetaGF&" }{TEXT -1 10 " " }{XPPEDIT 19 1 "z = rho *cos(phi);" "/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF&" } {TEXT -1 71 " \nThere are no restriction on the values of the spheric al coordinates." }}{PARA 15 "" 0 "" {TEXT -1 127 "rect2sph converts re ctangular coordinates to spherical coordinates using the formulas:\n \+ " }{XPPEDIT 19 1 "theta = arctan (y/x);" "/%&thetaG-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 28 " \+ in quadrants I and IV \n " }{XPPEDIT 19 1 "rho = sqrt(x^2 + y^2 + z ^2) ;" "/%$rhoG-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F+*$%\"zG\"\" #F+" }{TEXT -1 7 " " }{XPPEDIT 19 1 "theta = arctan(y/x) +Pi;" "i /%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F*%#PiGF*" }{TEXT -1 27 " in quadrant II " }{XPPEDIT 19 1 "phi = arccos(z/sqrt(x^2 \+ + y^2 + z^2));" "/%$phiG-%'arccosG6#*&%\"zG\"\"\"-%%sqrtG6#,(*$%\"xG\" \"#F)*$%\"yG\"\"#F)*$F(\"\"#F)!\"\"" }{TEXT -1 44 " \n \+ " }{XPPEDIT 19 1 "theta=arctan(y/x )-Pi;" "/ %&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F*%#PiGF," }{TEXT -1 84 " in quadrant III \nThe resulting spherical coordinates are restricted to the ranges " }{XPPEDIT 19 1 "rho >= 0;" "1\"\"!%$rhoG" }{TEXT -1 2 ", " }{XPPEDIT 19 1 " -Pi < theta; " "2,$%#PiG!\"\"%&thetaG" } {XPPEDIT 19 1 "``<= Pi;" "1%!G%#PiG" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "0 <= phi; " "1\"\"!%$phiG" }{XPPEDIT 19 1 "``<=Pi;" "1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 103 "sph2cyl converts s pherical coordinates to cylindrical coordinates using the formulas:\n \+ " }{XPPEDIT 19 1 "r = rho * sin(phi);" "/%\"rG*&%$rhoG \"\"\"-%$sinG6#%$phiGF&" }{TjEXT -1 12 " " }{XPPEDIT 19 1 "t heta = theta;" "/%&thetaGF#" }{TEXT -1 13 " " }{XPPEDIT 19 1 "z = rho *cos(phi);" "/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF&" } {TEXT -1 86 " \nThere are no restriction on the values of the spheric al or cylindrical coordinates." }}{PARA 15 "" 0 "" {TEXT -1 103 "cyl2s ph converts cylindrical coordinates to spherical coordinates using the formulas:\n " }{XPPEDIT 19 1 "rho = sqrt(r^2 + z^2) ; " "/%$rhoG-%%sqrtG6#,&*$%\"rG\"\"#\"\"\"*$%\"zG\"\"#F+" }{TEXT -1 9 " \+ " }{XPPEDIT 19 1 "theta = theta;" "/%&thetaGF#" }{TEXT -1 12 " " }{XPPEDIT 19 1 "phi = arccos(z/sqrt(r^2 + z^2));" "/%$ph iG-%'arccosG6#*&%\"zG\"\"\"-%%sqrtG6#,&*$%\"rG\"\"#F)*$F(\"\"#F)!\"\" " }{TEXT -1 140 " \nThere are no restriction on the values of the cy lindrical coordinates.\nThe resulting spherical coordinates are restri cted to the ranges " }{XPPEDIT 19 1 "rho >= 0;" "1\"\"!%$rhoG" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "0 <= phi; " "1\"\"!%$phiG" }k{XPPEDIT 19 1 "``<=Pi;" "1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 89 "These functions return floating point decimal numbers if the in put contains any decimals." }}{PARA 15 "" 0 "" {TEXT -1 289 "These fun ctions are part of the vec_calc package, and so can be used by name on ly after performing the command with(vec_calc) or with(vec_calc,functi on). The functions can always be accessed in the long forms vec_calc[ function]. The aliases can be used only after performing the command \+ " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cyl2rect([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"aG\"\"\"-%$cosG6#%\"bGF&*&F%F&-%$sinGF)F&% \"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2cyl([a,b,c]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,&*$%\"aG\"\"#\"\"\"*$%\"bGF(F )#F)F(-%'arctanG6$F+F'%\"lcG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sph2rect([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*(%\"aG\" \"\"-%$sinG6#%\"cGF&-%$cosG6#%\"bGF&*(F%F&F'F&-F(F-F&*&F%F&-F,F)F&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2sph([a,b,c]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,(*$%\"aG\"\"#\"\"\"*$%\"bGF(F)*$% \"cGF(F)#F)F(-%'arctanG6$F+F'-F06$*$,&F&F)F*F)F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sph2cyl([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"aG\"\"\"-%$sinG6#%\"cGF&%\"bG*&F%F&-%$cosGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "cyl2sph([a,b,c]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,&*$%\"aG\"\"#\"\"\"*$%\"cGF(F)#F) F(%\"bG-%'arctanG6$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texa s A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "CmoordConversion2D" 2 "CoordConversion2D" "" }{TEXT 26 2 ", " } {HYPERLNK 17 "vec_calc[deg2rad]" 2 "vec_calc[deg2rad]" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[rad2deg]" 2 "vec_calc[rad2deg]" "" } {TEXT -1 2 ". " }}}} yl([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"aG\"\"\"-%$sinG6#%\"cGF&%\"bG*&F%F&-%$cosGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "cyl2sph([a,b,c]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$,&*$%\"aG\"\"#\"\"\"*$%\"cGF(F)#F) F(%\"bG-%'arctanG6$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texa s A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "Cm<helpdimensionalcoordinatconversusingveccalcpackagfunctioncylrectconvertcylindricalrectangularsphsphericalaliascanusedafterexecutvccommandcallsequencthetarhophiparameterfirsthorizontalsecondverticalaxispositupwardrelatrighthandruleperpendiculardistancanglmeasurradiancounterclockwissamecautsystemmaplleftradialoriginpolardescriptformulanorestrictvaluquadrantiviiiiiresultrangarctanfunctwithargumentdesignproducexactneedthesreturnfloatpointdecimalnumberinputcontainanypartnameonlyperformalwayaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconversdegrad " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 13 "A 2D Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R:=makefunction(t,[t*cos(t),t*sin(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$:6#%\"tG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"-%$ cosG6#F.F/F)F):F'F)F*F)*&F.F/-%$sinGF2F/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([op(R(t)),t=-2*Pi..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0u 18 "curve_velocity(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$:6#%\"tG6\"6$%)operatorG%&arrowGF',&-%$cosG6#9$ \"\"\"*&F/F0-%$sinGF.F0!\"\"F'F':6#F&F'F(F',&F2F0*&F/F0F,F0F0F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "curve_acceleration(R);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$:6#%\"tG6\"6$%)operatorG%&arrowGF',& -%$sinG6#9$!\"#*&F/\"\"\"-%$cosGF.F2!\"\"F'F':6#F&F'F(F',&F3\"\"#*&F/F 2F,F2F5F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "curve_jerk(R ); #(The derivative of the acceleration.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$:6#%\"tG6\"6$%)operatorG%&arrowGF',&-%$cosG6#9$!\"$*& F/\"\"\"-%$sinGF.F2F2F'F':6#F&F'F(F',&F3F0*&F/F2F,F2!\"\"F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "curve_tangent(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$:6#%\"tG6\"6$%)operatorG%&arrowGF',$*&,&-% $cosG6#9$!\"\"*&F1\"\"\"-%$sinGF0F4F4F4,&F4F4*$F1\"\"#F4#F2F9F2F'F':F% F'F(F'*&F7F:,&F5F4*&F1F4F.F4F4F4F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_normal(R);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 $:6#%\v"tG6\"6$%)operatorG%&arrowGF',$*(,*-%$sinG6#9$\"\"#*&F.\"\"\"F1F 2F4*&F1F4-%$cosGF0F4F2*&F1\"\"$F6F4F4F4*&,&F2F4*$F1F2F4F2,&F4F4FF2F=F>F>F'F':F%F'F(F',$*(,*F6!\"#*&F6F4F1F2F>*&F1F4F.F4F2*&F1F9F. F4F4F4F:F?F=F>F>F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cur ve_curvature(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)ope ratorG%&arrowGF&*(,(*$9$\"\"#\"\"%*$F-F/\"\"\"F/F1F1,&F1F1F,F1!\"#*&,& F.F1F,F1F.F2!\"\"#F6F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L:=curve_arclength(R); L(0,2*Pi); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG:6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG 6$*$,&\"\"\"F2*$%\"tG\"\"#F2#F2F5/F4;9$9%F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$%\"tG\"\"#F(#F(F+/F*;\"\"!,$%#Pi GF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#PiG\"\"\",&F&F&*$F%\"\"# \"\"%#F&F)F&-%#lnG6#,&F%!\"#*$F'F+F&#!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "curve_tangential_acceleration(R);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)opewratorG%&arrowGF&*&9$\"\"\",&F, F,*$F+\"\"#F,#!\"\"F/F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "curve_normal_acceleration(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#: 6#%\"tG6\"6$%)operatorG%&arrowGF&*(,(*$9$\"\"#\"\"%*$F-F/\"\"\"F/F1F1, &F1F1F,F1!\"\"*&,&F.F1F,F1F.F2F3#F3F.F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "curve_forget(R);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A 3D Example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:= MF(t,[t*cos(t),t*sin(t),t]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG 7%:6#%\"tG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"-%$cosG6#F.F/F)F):F'F)F* F)*&F.F/-%$sinGF2F/F)F):F'F)F*F)F.F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "spacecurve([op(r(t)),t=-2*Pi..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Cv(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6\"6$%)operatorG%&arrowGF',&-%$cosG6#9$\"\"\"*&F/F0-%$ sinGF.F0!\"\"F'F':6#F&F'F(F',&F2F0*&F/F0F,F0F0F'F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ca(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%:6#x%\"tG6\"6$%)operatorG%&arrowGF',&-%$sinG6#9$!\"#*&F/\"\"\"-%$cosG F.F2!\"\"F'F':6#F&F'F(F',&F3\"\"#*&F/F2F,F2F5F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Cj(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6\"6$%)operatorG%&arrowGF',&-%$cosG6#9$!\"$*&F/\"\"\"- %$sinGF.F2F2F'F':6#F&F'F(F',&F3F0*&F/F2F,F2!\"\"F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CT(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6\"6$%)operatorG%&arrowGF',$*&,&-%$cosG6#9$!\"\"*&F1\" \"\"-%$sinGF0F4F4F4,&\"\"#F4*$F1F8F4#F2F8F2F'F':F%F'F(F'*&F7F:,&F5F4*& F1F4F.F4F4F4F'F':F%F'F(F'*$F7F:F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CN(r);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6 \"6$%)operatorG%&arrowGF',$*(,*-%$sinG6#9$\"\"%*&F.\"\"\"F1\"\"#F4*&F1 F4-%$cosGF0F4\"\"$*&F1F9F7F4F4F4*&,(*$F1F2F4*$F1F5\"\"&\"\")F4F4,&F5F4 F>F4!\"\"#FBF5FAFBFBF'F':F%F'F(F',$*(,*F7!\"%*&F7F4F1F5FB*&F1F4F.F4F9* &F1F9F.F4F4F4F;FCFAFBFBF'F':F%F'F(F',$*(F;FCF1F4FAFBFBF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CB(r);" y}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6\"6$%)operatorG%&arrowGF'*&,(*$9$\"\"%\"\"\"*$F.\"\"# \"\"&\"\")F0#!\"\"F2,&-%$cosG6#F.!\"#*&F.F0-%$sinGF:F0F0F0F'F':F%F'F(F ',$*&,&F=F2*&F.F0F8F0F0F0F,F5F6F'F':F%F'F(F'*&,&F2F0F1F0F0F,F5F'F'" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ck(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)operatorG%&arrowGF&*(,(*$9$\"\"%\"\"\"*$ F-\"\"#\"\"&\"\")F/F/,&F1F/F0F/!\"#*&F+F/F4!\"\"#F7F1F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ct(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)operatorG%&arrowGF&*&,&*$9$\"\"#\"\"\"\"\"'F/F/,( *$F-\"\"%F/F,\"\"&\"\")F/!\"\"F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "L:=CL(r); L(0,2*Pi); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG:6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG 6$*$,&\"\"#\"\"\"*$%\"tGF2F3#F3F2/F5;9$9%F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"#\"\"\"*$%\"tGF(F)#F)F(/F+;\"\"!,$%#Pi GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#PiG\"\"\",&\"\"#F&*$F%F( \"\"%#F&F(zF&-%#lnG6#,&*&F(F+F%F&F&*$,&F&F&F)F(F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaT(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #:6#%\"tG6\"6$%)operatorG%&arrowGF&*&9$\"\"\",&\"\"#F,*$F+F.F,#!\"\"F. F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaN(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)operatorG%&arrowGF&*(,(*$9$\"\"% \"\"\"*$F-\"\"#\"\"&\"\")F/F/,&F1F/F0F/!\"\"*&F+F/F4F5#F5F1F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_forget(r);" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Bel monte and Philip B. Yasskin\n Department of Mathematics, Texas A& M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " \+ " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "curve_forget" 2 "curve_forget" "" }{TEXT -1 1 "." }}}}  (indexedJ رCE (function (listJ3F`E {(set (stringJE}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaT(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #:6#%\"tG6\"6$%)operatorG%&arrowGF&*&9$\"\"\",&\"\"#F,*$F+F.F,#!\"\"F. F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaN(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"tG6\"6$%)operatorG%&arrowGF&*(,(*$9$\"\"% \"\"\"*$F-\"\"#\"\"&\"\")F/F/,&F1F/F0F/!\"\"*&F+F/F4F5#F5F1F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_forget(r);" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Bel monte 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 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "curve_forget" 2 "curve_forget" "" }{TEXT -1 1 "." }}}}  (indexedJ رCE (function (listJ3F`E {<helpfrenetanalysicurvusingveccalcpackagcallsequenccommandparameterlistbelowaliaformarrowdefinfunctionparametdescriptthesdesignperformeachshortmostworkanydimenshowevbinormaltorsonlycorrespondactionvelocitcvcalculatacceleratcajerkcjtangctunitnormalcnprincipalcbcurvaturckarclengthclarclengthtangentialcatcanforgetcforgetclearremembtablabovusenameyoumustfirstexecutwithvcaliasproducmakefunctmfsuchmayplottplotparametricargumspacecurvusesspeedupcomputatafterfinishavoidcluttermemordoneexamplcossinoppiderivatvalucopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocn<<<collect<command3C<<<<<<<<<<<<complex << component<comput <<consist<contain<<<<convers <<convert<<<<< coordconvers<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:"~ }{TEXT -1 1 " \+ " }{TEXT 259 25 "vec-calc[curve_forget] - " }{TEXT -1 42 "Clears Remem ber 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 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT 257 27 " Cforget = curve_forget" }}{PARA 0 " " 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 "\n" }{TEXT 258 74 " curve_forget(r,s...) Cforget(r,s...) vec_calc[curve_forget] (r,s...)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 14 " r,s... - " }{TEXT -1 93 "a sequence of curves, e ach in the form of a list of arrow-defined functions of one parameter. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " \+ " }}{PARA 15 "" 0 "" {TEXT -1 50 "The commands in the vec_calc package to perform a " }{HYPERLNK 17 "Frenet analysis of a curve" 2 "Curve" " " }{TEXT -1 310 " (such as curve_velocity and curve_acceleration) use \+ remember tables to store their results. This cuts down on computing t ime for other commands. The command curve_forget(r) clears these reme mber tables for the curve r. The command curve_forget(r,s...) clears \+ these remember tables for all the curves r,s..." }}{PARA 15 "" 0 "" {TEXT -1 314 "This command is part of the vec_calc package, and so can be used in the form curve_forget only after performing the command wi th(vec_calc) or with(vec_calc, curve_forget). The command can always \+ be accessed in the long form vec_calc[curve_forget]. The alias Cforge t can be used only after performing the command " }{HYPERLNK 17 "vc_al iases" 2 "vc_aliases" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r:=makefunction(t,[t,sin(t),cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%:6#%\"tG6\"6$%)operatorG%&arrowGF)9$F)F)%$ sinG%$cosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CB(r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6#%\"tG6\"6$%)operatorG%&arrowGF',$ *$\"\"##\"\"\"F-#!\"\"F-F'F':F%F'F(F',$*&F-F.-%$cosG6#9$F/F.F'F':F%F'F (F',$*&F-F.-%$sinGF7F/F0F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cforget(r); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyrig ht 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department \+ of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "S ee Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 1 "." }}}} 6 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r:=makefunction(t,[t,sin(t),cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%:6#%\"tG6\"6$%)operatorG%&arrowGF)9$F)F)%$ sinG%$cosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 bndcritpt<both<bottom<boundar<boxed<bracket<but << byperform<ca<<<calc{1< << <<< < << < <<< << < < < < < << < < < < = = = = calculat_3<<<<<< <<<<<<<<<<<<<====calculu <<callw<<<<<<<<<<<<<<<<<<<<<<<<<====can{X<<<<<<<<<<<<<<<<<<<<<<<<<<====cat<<<caut<<<cb<<<<cforget <<<<chang<choic<<<<<<<cj<<<ck<<< Q RETURNJCh}  Q cl<<<classif<clear <<click<clutter<cn<<<collect<command{<<<<<<<<<<<<< <<<<<<<<<<<<<====complex <<compon< component<comput+ <<<<<<====computat<conclus<confirm<consist< constrain< constraint<contain<<<<<<<contour< contourplot<convers <<convert<<<<< < coordconvers <<<<<< coordinat#;<<< < <<<<,vzJ@vvJ,v@zJwwyTyyTyJLw%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"%\"bG\matric<matrix<<<<<<max<<<<<<maxima<maximum <<may<<<<measur<<<memor<method<mf+<<<<<<====min<<<<<<minima<minimum <<minor&<<<<<<miss< modificat <<more <<most<mtaylor<muint'<<<<< ====mulitipl<multi<<<<<multidimensional<multipl< multipleint<<< multipli< multivariabl <<must#<<<<<<==namec&<<<<<<<<<<<<<<<<<<<<====0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 <functveccalccurvforgetclearremembtablanalysialiacanusedafterexecutvcaliascommandcforgetcallsequencparametereachformlistarrowdefinfunctionparametdescriptpackagperformfrenetsuchvelocitacceleratusestortheirresultcutsdowncomputtimeotherthesallpartonlywithalwayaccesslongexamplmakefunctsincoscbcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoormatequalreturnoptionaldescriptdesignwithdefinitdifferentiatantiincludparametrsurfaceachownpageexampluseyoufirstcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvmultimaxminity " }}{PARA 0 "" 0 "" {TEXT 26 9 "S ee Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 1 "." }}}} 6 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0er" 1 10 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 }{CSTYLE "" -1 268 "Courier" 1 10 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 }{CSTYLE "" -1 270 "Courier" 1 10 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 }{CSTYLE " " -1 272 "Courier" 1 10 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 }{CSTYLE "" -1 274 "Courier" 1 10 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 }{CSTYLE "" -1 276 "Courier" 1 10 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 }{CSTYLE "" -1 278 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "Time s" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "Time s" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 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 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 42 " \+ Multivariable Max-Min Problems using the " }{TEXT 256 8 "vec_calc" } {TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT 26 9 "Functions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT 277 3 " " }{HYPERLNK 17 "vec_cal c[GRAD]" 2 "vec_calc[GRAD]" "" }{TEXT 278 9 " " }{TEXT 290 4 " - " }{TEXT 279 22 "Calculate the Gradient" }}{PARA 261 "" 0 "" {TEXT 259 3 " " }{HYPERLNK 17 "student[equate]" 2 "vec_calc[GRAD]" " " }{TEXT 260 12 " - " }{TEXT 280 74 "Set the Gradient equal t o zero or set up the Lagrange Multiplier equations" }}{PARA 257 "" 0 " " {TEXT 261 3 " " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT 262 22 " - " }{TEXT 281 31 "Solve for exact critical points " }}{PARA 258 "" 0 "" {TEXT 263 3 " " }{HYPERLNK 17 "fsolve" 2 "fsol ve" "" }{TEXT 264 21 " - " }{TEXT 282 45 "Solve for \+ approximate decimal critical points" }}{PARA 259 "" 0 "" {TEXT 265 3 " " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT 266 18 " \+ - " }{TEXT 283 33 "Evaluate solutions which contain " } {HYPERLNK 17 "RootOf" 2 "RootOf" "" }{TEXT 284 2 "'s" }}{PARA 260 "" 0 "" {TEXT 267 3 " " }{HYPERLNK 17 "vec_calc[HESS]" 2 "vec_calc[HESS ]" "" }{TEXT 268 13 " - " }{TEXT 285 21 "Calculate the Hessi an" }}{PARA 262 "" 0 "" {TEXT 269 3 " " }{HYPERLNK 17 "vec_calc[lead ing_principal_minor_determinants]" 2 "vec_calc[leading_principal_minor _determinants]" "" }{TEXT 270 5 " - " }{TEXT 286 86 "Calculate the L eading Principal Minor Determinants to apply the Second Derivative Tes t" }}{PARA 263 "" 0 "" {TEXT 271 3 " " }{HYPERLNK 17 "subs" 2 "subs " "" }{TEXT 272 23 " - " }{TEXT 287 49 "Convert a \+ solution set into a list of coordinates" }}{PARA 264 "" 0 "" {TEXT 273 3 " " }{HYPERLNK 17 "op" 2 "op" "" }{TEXT 274 25 " \+ - " }{TEXT 288 31 "Strip brackets of a list or set" }}{PARA 265 "" 0 "" {TEXT 275 3 " " }{HYPERLNK 17 "vec_calc[makefunction]" 2 "vec_calc[makefunction]" "" }{TEXT 276 5 " - " }{TEXT 289 30 "Make an arrow-defined function" }}{PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" } {TEXT -1 52 " - These aliases can be used after execution of the " } {HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 9 " command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 48 " LPMD \+ = leading_principal_minor_determinants" }}{PARA 0 "" 0 "" {TEXT 258 24 " MF = makefunction" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 " Description:" }}{PARA 15 "" 0 "" {TEXT -1 147 "These commands are desi gned to help with multidimensional max-min problems. Below are exampl es of both the unconstrained and constrained problems." }}{PARA 15 "" 0 "" {TEXT -1 363 "To use the command names, GRAD, HESS, leading_princ ipal_minor_determinants or makefunction, you must first execute with(v ec_calc) or with(vec_calc,GRAD, HESS, leading_principal_minor_determin ants, makefunction).\nTo use the command name, equate, you must first \+ execute with(student) or with(student, equate).\nTo use the aliases, y ou must first execute the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_a liases" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Exa mples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): v c_aliases:" }}}{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#>%\"fG:6$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$F.\"\"##!\"\"F6*$F0F6#F8\"\" )F/F)F)" }}}{PARA 5 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 " " {TEXT -1 83 "Compute the gradient of f, set it equal to zero and sol ve for the critical points: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%delfG7$:6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"-%$expG6#,&*$9$\"\"##!\" \"F8*$F0F8#F:\"\")F1F1*(F7F8F0F1F2F1F:F*F*:F'F*F+F*,&*&F7F1F2F1F1*(F7F 1F0F8F2F1#F:\"\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq s:=equate(delf(x,y),[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG <$/,&*&%\"yG\"\"\"-%$expG6#,&*$%\"xG\"\"##!\"\"F1*$F)F1#F3\"\")F*F**(F 0F1F)F*F+F*F3\"\"!/,&*&F0F*F+F*F**(F0F*F)F1F+F*#F3\"\"%F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "critpts:=solve(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(critpt s[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 maximum, a minimum or a saddle point. Note, the \+ test may fail. First, compute the Hessian and the leading principal m inor determinants: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Hf:=H ESS(f): matrix(Hf(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRI XG6#7$7$,&*(%\"xG\"\"\"%\"yGF+-%$expG6#,&*$F*\"\"##!\"\"F2*$F,F2#F4\" \")F+!\"$*(F*\"\"$F,F+F-F+F+,*F-F+*&F*F2F-F+F4*&F,F2F-F+#F4\"\"%*(F*F2 F,F2F-F+#F+F?7$F;,&F)#F8F?*(F*F+F,F:F-F+#F+\"#;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "leading_principal_minor_determinants(Hf(x,y)): " }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determinants: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",&*(%\"xGF'%\"yGF'- %$expG6#,&*$F*\"\"##!\"\"F1*$F+F1#F3\"\")F'!\"$*(F*\"\"$F+F'F,F'F'" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,2*(%\"xGF'%\"yGF'-%$exp G6#,&*$F*F'#!\"\"F'*$F+F'#F2\"\")F'#\"\"&\"\"%*(F*F'F+F8F,F'#F2\"#;*(F *F8F+F'F,F'#F2F8*$F,F'F2*&F*F'F,F'F'*&F+F'F,F'#\"\"\"F'*&F*F8F,F'F2*&F +F8F,F'F:" }}}{PARA 0 "" 0 "" {TEXT -1 116 "At each critical point, ev aluate the Hessian and the leading principal minor determinants and in terpret the results:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p1' =p1; H1:=Hf(op(p1)); LPMD(H1):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%#p1G7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7$\"\"!\"\" \"7$F(F'" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Deter minants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 73 "Since D[2] is negative, the first critical point (0,0) \+ is a saddle point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p2'=p 2; H2:=Hf(op(p2)); LPMD(H2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% #p2G7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7$7$,$-%$ex pG6#!\"\"!\"%\"\"!7$F-,$F(F+" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading P rincipal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"D G6#\"\"\",$-%$expG6#!\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" DG6#\"\"#,$*$-%$expG6#!\"\"F'\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Si nce D[2] is positive and D[1] is negative, the second critical point ( 1,2) is a local maximum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " 'p3'=p3; H3:=Hf(op(p3)); LPMD(H3):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p3G7$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G7$7$ ,$-%$expG6#!\"\"\"\"%\"\"!7$F-F(" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leadi ng Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"DG6#\"\"\",$-%$expG6#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"DG6#\"\"#,$*$-%$expG6#!\"\"F'\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Since D[2] is positive and D[1] is positive, the third critical po int (1,-2) is a local minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p4'=p4; H4:=Hf(op(p4)); LPMD(H4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p4G7$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#H4G7$7$,$-%$expG6#!\"\"\"\"%\"\"!7$F-F(" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$expG6#!\"\"\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$-%$expG6#!\"\"F'\"\"%" }}}{PARA 0 " " 0 "" {TEXT -1 100 "Since D[2] is positive and D[1] is positive, the \+ fourth critical point (-1,2) is a local minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p5'=p5; H5:=Hf(op(p5)); LPMD(H5):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p5G7$!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H5G7$7$,$-%$expG6#!\"\"!\"%\"\"!7$F-,$F(F+" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$expG6#!\"\"!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$-%$expG6#!\"\"F'\"\" %" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Since D[2] is positive and D[1] is negative, the fifth critical point (-1,-2) is a local maximum." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "To confir m the conclusions, use a contour plot: (Try rotating the plot.)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "contourplot3d(f(x,y), x=-2.. 2, y=-3..3, orientation=[-90,0], axes=boxed); " }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 24 "The Constrained Problem:" }}{PARA 0 "" 0 "" {TEXT -1 97 "Find the absolute maximum and minimum values of a function insi de or on the boundary of a region." }}{PARA 5 "" 0 "" {TEXT -1 8 "Exam ple:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Extremize the function:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=(x,y)->x*y*exp(-x^2/2-y^2 /8); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$F.\"\"##!\"\"F6*$F0F6#F8\" \")F/F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 39 "inside or on the ellipse g( x,y)=1 where" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=(x,y)->x^ 2/4 + y^2/16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF),&*$9$\"\"##\"\"\"\"\"%*$9%F0#F2\"#;F)F)" }}} {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 142 "The interior critical points were found in the unconstrained example. There are three methods of finding the critical points on the bounda ry." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 42 "Boundary Method I: Elimi nate a Variable " }}{PARA 0 "" 0 "" {TEXT -1 38 "Solve the constraint \+ for one variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y0:=sol ve(g(x,y)=1,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G6$,$*$,&*$%\" xG\"\"#!\"\"\"\"%\"\"\"#F.F+F+,$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 212 "Notice that we named the solution y0 instead of y so that we can \+ still use y as a variable. Also notice that there are 2 solutions for the upper and lower halves of the ellipse. We must handle these sepa rately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the upper half of the boundary, substitute the solution into t he function and differentiate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f1:=makefunction(x,f(x,y0[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G:6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(9$\"\"\",&*$F.\"\"#!\" \"\"\"%F/#F/F2-%$expG6#!\"#F/F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Df1:=D(f1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df1 G:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&,&*$9$\"\"#!\"\"\"\"%\"\"\"#F4F 1-%$expG6#!\"#F4F1*(F0F1F.#F2F1F6F4F9F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 90 "Find the x-coordinate at each critical point and substitute bac k to find the y-coordinate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df1(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$ *$\"\"##\"\"\"F',$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'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$\"\"##\"\"\"F',$F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y2:=subs(x=x0[2],y0[1]); b2:=[x0[2],y2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*$\"\"##\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$,$*$\"\"##\"\"\"F(!\"\",$F'F(" }}} {PARA 0 "" 0 "" {TEXT -1 42 "Repeat for the lower half of the boundary :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f2:=makefunction(x,f(x, y0[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G:6#%\"xG6\"6$%)opera torG%&arrowGF(,$*(9$\"\"\",&*$F.\"\"#!\"\"\"\"%F/#F/F2-%$expG6#!\"#F/F 9F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Df2:=D(f2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df2G:6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&,&*$9$\"\"#!\"\"\"\"%\"\"\"#F4F1-%$expG6#!\"#F4F9*(F0F1F.#F2F1F 6F4F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df2( x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$\"\"##\"\"\"F',$ 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'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$*$\"\"## \"\"\"F',$F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y4:=sub s(x=x0[2],y0[2]); b4:=[x0[2],y4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#y4G,$*$\"\"##\"\"\"F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7 $,$*$\"\"##\"\"\"F(!\"\",$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 139 "Fin ally, we tabulate the values of the function at all interior and bound ary critical points and identify the absolute maximum and minimum:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p1; f(op(p1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p2; f(op(p2)); evalf (\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p3; f(op(p3)); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$ \"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"\"!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p4; f(op(p4)); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$ex pG6#!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+C))edt!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p5; f(op(p5)); evalf(\");" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1; f(op (b1)); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$\"\"##\"\"\"F %,$F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2; f(op(b2)); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$\"\"##\"\"\"F&!\"\",$F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $!+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b3; f(op(b3) ); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$\"\"##\"\"\"F%,$F $!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b4; f(op(b4)); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$\"\"##\"\"\"F&!\"\",$F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+G8T8a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 154 "So we see that the abs olute maxima occur at the interior points (1,2) and (-1,-2), and the a bsolute minima occur at the interior points (1,-2) and (-1,2)." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 47 "Boundary Method II: Parametrize the Boundary " }}{PARA 0 "" 0 "" {TEXT -1 27 "Define the parametrizat ion:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r:=makefunction(t,[2 *cos(t),4*sin(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7$:6#%\"t G6\"6$%)operatorG%&arrowGF),$-%$cosG6#9$\"\"#F)F):F'F)F*F),$-%$sinGF0 \"\"%F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Restrict the function to th e boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fr:=makefunct ion(t,simplify(f(op(r(t)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fr G:6#%\"tG6\"6$%)operatorG%&arrowGF(,$*(-%$cosG6#9$\"\"\"-%$sinGF0F2-%$ expG6#!\"#F2\"\")F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Find the critic al points on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Dfr:=D(fr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DfrG:6#%\"tG6\"6 $%)operatorG%&arrowGF(,&*&-%$sinG6#9$\"\"#-%$expG6#!\"#\"\"\"!\")*&-%$ cosGF0F2F3F7\"\")F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bn dcritpts:=solve(Dfr(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+bnd critptsG6$,$%#PiG#\"\"\"\"\"%,$F'#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "b1:=r(bndcritpts[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$\"\"##\"\"\"F',$F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b2:=r(bndcritpts[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$\"\"##\"\"\"F',$F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Since the equation is non-polynomial, solve may not give all solu tions. So we plot the function for one period:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "plot(Dfr,-Pi..Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 85 "From the plot and its symmetries, it is obvious that solve miss ed two more solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b3 :=r(3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$,$*$\"\"##\"\" \"F(!\"\",$F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b4:=r(-3 *Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$\"\"##\"\"\"F(! \"\",$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 206 "Finally, we tabulate th e values of the function at all interior and boundary critical points \+ and identify the absolute maximum and minimum: \n(This was done with t he first method. So we won't redo it here.)" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 "Boundary Method III: Lagrange Multiplier Method" }} {PARA 0 "" 0 "" {TEXT -1 68 "Find the gradient of thefunction and the \+ gradient of the constraint:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%delfG7$:6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"-%$expG6#,&*$9$\"\"##!\" \"F8*$F0F8#F:\"\")F1F1*(F7F8F0F1F2F1F:F*F*:F'F*F+F*,&*&F7F1F2F1F1*(F7F 1F0F8F2F1#F:\"\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "de lg:=GRAD(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG7$:6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF*,$9$#\"\"\"\"\"#F*F*:F'F*F+F*,$9%#F1\"\")F* F*" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Set up the Lagrange multiplier equ ations and solve for the critical points on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqs:=equate(delf(x,y),lambda*delg(x ,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,&*&%\"yG\"\"\"-%$e xpG6#,&*$%\"xG\"\"##!\"\"F1*$F)F1#F3\"\")F*F**(F0F1F)F*F+F*F3,$*&%'lam bdaGF*F0F*#F*F1/,&*&F0F*F+F*F**(F0F*F)F1F+F*#F3\"\"%,$*&F:F*F)F*#F*F6 " }}}{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#!\"#!\"%/%\"xG-%'RootOfG6#,&*$%#_ZG\"\"#\"\"\"F -F8/%\"yG,$F1F7<%/F(,$F*\"\"%/F0,$F1!\"\"F9" }}}{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#!\"#!\"%/%\"xG*$\"\"##\"\"\"F2/%\"yG,$F1F2<%F'/F0,$F1!\"\"/F6,$F 1F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1:=subs(sol1[1],[x, y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$\"\"##\"\"\"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'!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol2:=allvalues(s ol[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G6$<%/%'lambdaG,$-%$ expG6#!\"#\"\"%/%\"yG,$*$\"\"##\"\"\"F3F3/%\"xG,$F2!\"\"<%F'/F7F2/F0,$ F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b3:=subs(sol2[1],[x ,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$,$*$\"\"##\"\"\"F(!\" \",$F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2:=subs(sol2[2 ],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$\"\"##\"\"\"F', $F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 208 "Finally, we tabulate the val ues of the function at all interior and boundary critical points and i dentify the absolute maximum and minimum: \n (This was done with the \+ first method. So we won't redo it here.)" }}}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 115 "- Copyright 1995-97 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 " vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "GRAD" 2 "GRA D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "equate" 2 "equate" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "leading_principal_minor_determinants" 2 "leading_principal_minor_determinants" "" }{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 ". " }}}} "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "GRAD" 2 "GRA D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "equate" 2 "equate" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "leading_principal_minor_determinants" 2 "leading_principal_minor_determinants" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "op" 2 "op" "" }{TEXT -1 2 ", " }{HYPERLNK 17problem <<produc<<<<product<<<proper<publish<quadrant <<quot <<rad<<<<<radial <<radian<<<radical<ramp ==rang<<<====real<rect6< << < rectangular << recurrsive<redo<referenc<region<relat<releas<rememb <<repeat< represent<<< requir#<<<<<<<<reserv<resolv<respect<restrict <<<result<<<<<return?<<<<<<<<<<<<<<<rho<<<right<<<rng====root<rootof<rotat<rule<saddl<same<<<<scalar3S<<<<<<<<====second'<<<<<====select<separate<sequencw<<<<<<<<<<<<<<<<<<<<<<<<<====set<sets<several<<<< 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 15 "vec_calc[ss] - " }{TEXT -1 41 "Symbolifound<fourth<==fr<frenet<<<fsolv<functso<< <<<<<<<<<<< < <<<< <<<<<<====function[4<<<<<<<<<<<<<<<<<<====gett<give<given <<grad'"<<<<<<<<<gradi <<<<group<half<halv<hand<handl<have<having<help< <<<<<here <<hess<<<<<< <J ߺJ  J4 DJ hJ J J0 J Jp $Jd HdefinS"<<<<<<<<<<<<<<<<====definit<<<deg<<<<<degre<delf <<delg <<densit<departm{<<<<<<<<<<<<<<<<<<<<<<<<<<====derivat <<<<<descript{<<<<<<<<<<<<<<<<<<<<<<<<<<====design<<<<<<det<<<<< <determin<<< determinant*<<<<<<<uevalb uevalfJm@ uevalhfJOrpYG0Ql0 uevalnJu uexpand 쥼JyfvPjvLQЧ ufrontendJG4identifevalfmaximaoccurminimaiiparametrizparametrizatcossinrestrictfrsimplifdfrbndcritptnonpolynomialgiveperiodpisymmetrobvioumissmoredonewonredohereiiithefunctdelglambdasolinvolvresolvthemcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitderivattestsubsconvertsolutintolistcoordinatopstripbracketmakefunctmakearrowdefinfunctaliasthescanusedafterexecutvccommandlpmdmfdescriptdesignwithmultidimensionalbelowexamplbothunconstrainconstrainusenameyoumustfirstfindallclassifeachlocalmaximumminimumsaddlexpcomputdelfeqscritptdeterminnotemayfailhfmatrixatinterpretresultsincnegatpositthirdfourthfifthconfirmconcluscontourplottryrotatcontourplotorientataxesboxedabsolutvaluinsidboundarregionextremizellipsinteriorwerefoundmethodeliminatvariablconstraintnoticwenamedinsteadstillalsoupperlowerhalvhandlseparatehalfsubstitutdifferentiatdfbackrepeatfinaltabulat<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 17 "vec_calc[GRAD] - " }{TEXT -1 56 "Calculates the Gradi ent of a Function in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 20 "C alling Sequences: \n" }{TEXT 256 41 " GRAD(fcn) vec_calc[GR AD](fcn)" }}{PARA 0 "" 0 "" {TEXT 258 46 " GRAD(fcn,vars) vec_ca lc[GRAD](fcn,vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 12 " fcn - " }{TEXT -1 57 "a scalar function of several variables in arrow notation " }} {PARA 0 "" 0 "" {TEXT 259 12 " vars - " }{TEXT -1 72 "a list of na mes to be used as the independent variables of the function " }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 86 "The gradient of a function is the vector of first partial derivatives of the function." }}{PARA 15 "" 0 "" {TEXT -1 156 "GRAD acts on an arrow-defined function and returns a lis t of the first partial derivatives as arrow-defined functions. The ch oice of variables is optional." }}{PARA 15 "" 0 "" {TEXT -1 30 "GRAD d iffers from the command " }{HYPERLNK 17 "grad" 2 "grad" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 131 " pack age:\nlinalg[grad] acts on an expression and returns a vector of expre ssions. It requires the specification of the variables. " }}{PARA 15 "" 0 "" {TEXT -1 230 "The function GRAD is part of the vec_calc packag e, and so can be used in the form GRAD only after performing the comma nd with(vec_calc) or with(vec_calc, GRAD). The function can always be accessed in the long form vec_calc[GRAD]." }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{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:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF**(9$ \"\"#9%\"\"$9&\"\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " delf:=GRAD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delfG7%:6%%\"xG% \"yG%\"zG6\"6$%)operatorG%&arrowGF+,$*(9$\"\"\"9%\"\"$9&\"\"%\"\"#F+F+ :F'F+F,F+,$*(F1F7F3F7F5F6F4F+F+:F'F+F,F+,$*(F1F7F3F4F5F4F6F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "delf(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(%\"xG\"\"\"%\"yG\"\"$%\"zG\"\"%\"\"#,$*(F&F, F(F,F*F+F),$*(F&F,F(F)F*F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=makefunction([x,y],2*x^2*y+exp(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&9$\"\" #9%\"\"\"F0-%$expG6#F1F2F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "delg:=GRAD(g,[a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG 7$:6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF*,$*&9$\"\"\"9%F1\"\"%F*F*:F'F *F+F*,&*$F0\"\"#F7-%$expG6#F2F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "delg(p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&% \"pG\"\"\"%\"qGF'\"\"%,&*$F&\"\"#F,-%$expG6#F(F'" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[grad ]" 2 "linalg[grad]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Dif fops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_M in" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DIV" 2 "DIV" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HES S" 2 "HESS" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 2 ". " }}}} @@h (Multi_Max_Minhalf<H 20 "6#>%%delgG 7$:6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF*,$*&9$\"\"\"9%F1\"\"%F*F*:F'F *F+F*,&*$F0\"\"#F7-%$expG6#F2F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "delg(p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&% \"pG\"\"\"%\"qGF'\"\"%,&*$F&\"\"#F,-%$expG6#F(F'" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmo<~{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 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 }{CSTYLE " " -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" <{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 16 "vec_calc[DIV] - " }{TEXT -1 62 "Calculates the Diverg ence of a Vector Field in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 35 " DIV(F) vec_calc[DIV](F)" }}{PARA 0 "" 0 "" {TEXT 258 40 " DIV(F,vars) vec_calc[DIV](F,vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 12 " \+ F - " }{TEXT -1 96 "an n-dimensional vector field in the form of a list of n arrow-defined functions of n variables " }}{PARA 0 "" 0 " " {TEXT 259 12 " vars - " }{TEXT -1 58 "a list of n names to be us ed as the independent variables " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 139 "DIV returns the divergence of a vector field in the form of an arrow-defi ned function of n variables. The choice of variables is optional." }} {PARA 15 "" 0 "" {TEXT -1 29 "DIV differs from the command " } {HYPERLNK 17 "diverge" 2 "diverge" "" }{TEXT -1 8 " in the " } {HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 136 " package:\nlinalg[d iverge] acts on a vector of n expressions and returns an expression. \+ It requires the specification of the variables. " }}{PARA 15 "" 0 "" {TEXT -1 226 "The function DIV is part of the vec_calc package, and so can be used in the form DIV only after performing the command with(ve c_calc) or with(vec_calc, DIV). The function can always be accessed i n the long form vec_calc[DIV]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "F:=make function([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%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG F+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8 F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "DIV(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%\"xG%\"yG%\"zG6\"6 $%)operatorG%&arrowGF(,*9$\"\"#*&9%\"\"\"9&\"\"%F1*&F0F1F2F1F.*&F-F.F2 F.\"\"$F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995 -97 by Arthur Belmonte and Philip B. Yasskin\n Department of Math ematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also :" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[diverge]" 2 "linalg[diverge]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VEC_POT" 2 "VEC_POT" "" } {TEXT -1 2 ". " }}}} etermin< determinant<<<<^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG F+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8 F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MP<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 17 "vec_calc[CURL] - " }{TEXT -1 74 "Calculates the Curl \+ of a Three-Dimensional Vector Field in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT 256 37 " CURL(F) vec_calc[CURL](F)" }}{PARA 0 "" 0 "" {TEXT 258 42 " CURL(F,vars) vec_calc[CURL](F,vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 12 " F - " }{TEXT -1 95 "a 3-dimensional vector field in the form of a list of 3 arrow-defined functions of 3 variables " } }{PARA 0 "" 0 "" {TEXT 259 12 " vars - " }{TEXT -1 58 "a list of 3 names to be used as the independent variables " }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 144 "CURL returns the curl of a vector field in the form of a list of 3 arrow-defined functions of 3 variables. The choice of vari ables is optional." }}{PARA 15 "" 0 "" {TEXT -1 30 "CURL differs from \+ the command " }{HYPERLNK 17 "curl" 2 "curl" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 145 " package:\nlinalg[ curl] acts on a vector of 3 expressions and returns a vector of 3 expr essions. It requires the specification of the variables. " }}{PARA 15 "" 0 "" {TEXT -1 230 "The function CURL is part of the vec_calc pac kage, and so can be used in the form CURL only after performing the co mmand with(vec_calc) or with(vec_calc, CURL). The function can always be accessed in the long form vec_calc[CURL]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc);" }}}{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%:6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"% F3F+F+:F'F+F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CURL(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF),$*$9%\"\"#!\"\"F)F):F %F)F*F),&*(9$\"\"\"F/F69&\"\"$\"\"%*&F5F6F7F8!\"#F)F):F%F)F*F),&F.!\"$ *&F5F6F7F9F1F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "linalg[curl]" 2 "linalg[curl]" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DIV " 2 "DIV" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VEC_POT" 2 "VEC_POT" "" }{TEXT -1 2 ". " }}}} New@O1 vIh6San+,(*$9$\"\"#\"\"\"*$9%\"\"$F3*(F1F3F5F39&\"\"% F3F+F+:F'F+F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+" }}<functveccalcgradcalculatgradiarrownotatcallsequencfcnvarsparameterscalarseveralvariabllistnameusedindependdescriptvectorfirstpartialderivatactsdefinreturnfunctionchoicoptionaldiffercommandlinalgpackagexpressrequirspecificatpartcanformonlyafterperformwithalwayaccesslongexampldelfmakefunctexpdelgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopmultimaxmindivcurlhesspot<functveccalcdivcalculatdivergencvectorfieldarrownotatcallsequencvarsparameterdimensionalformlistdefinfunctionvariablnameusedindependdescriptreturnchoicoptionaldiffercommanddiverglinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopgradcurlpotrect<independ<<<input <<insid<instead <<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "MS Sans Serif" 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "MS Sans Serif" 0 1 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 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 263 16 "vec_calc[LAP] - " }{TEXT -1 78 "Calculates the Laplac ian of a Function or List of Functions in Arrow Notation " }}{PARA 0 " " 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 39 " LAP(fcn) vec_calc[LAP](fcn)" }}{PARA 0 "" 0 "" {TEXT 257 44 " LAP(fcn,vars) vec_calc[LAP](fcn,vars)" }} {PARA 0 "" 0 "" {TEXT 259 37 " LAP(F) vec_calc[LAP](F)" } }{PARA 0 "" 0 "" {TEXT 260 42 " LAP(F,vars) vec_calc[LAP](F,va rs)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 261 12 " fcn - " }{TEXT -1 42 "an arrow-de fined function of n variables " }}{PARA 0 "" 0 "" {TEXT 262 12 " F \+ - " }{TEXT -1 58 "a list or array of arrow-defined functions of n \+ variables " }}{PARA 0 "" 0 "" {TEXT 258 12 " vars - " }{TEXT -1 58 "a list of n names to be used as the independent variables " }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 174 "If the argument is a single function, L AP returns the Laplacian of the function in the form of an arrow-defin ed function of n variables. The choice of variables is optional." }} {PARA 15 "" 0 "" {TEXT -1 96 "If the argument is a list or array, LAP \+ recurrsively maps onto the entries in the list or array." }}{PARA 15 " " 0 "" {TEXT -1 29 "LAP differs from the command " }{HYPERLNK 17 "lapl acian" 2 "laplacian" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 126 " package:\nlinalg[laplacian] acts on an e xpression and returns an expression. It requires the specification of the variables. " }}{PARA 15 "" 0 "" {TEXT -1 226 "The function LAP is part of the vec_calc package, and so can be used in the form LAP only after performing the command with(vec_calc) or with(vec_calc, LAP). \+ The function can always be accessed in the long form vec_calc[LAP]." } }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{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%: 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$9$\"\"#\"\"\"*$9%\"\"$F 3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "LAP(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF),(\" \"#\"\"\"9%\"\"'*(9$F/F0F/9&F.\"#7F)F):F%F)F*F),$F4F.F)F):F%F)F*F),&*$ F4\"\"$F.*&F3F.F4F/F1F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- \+ Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Depa rtment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[laplacian]" 2 "linalg[lapl acian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "DIV" 2 "DIV" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" }{TEXT -1 2 ". " }}}} hFG7%: 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$9$\"\"#\"\"\"*$9%\"\"$F 3*(F1F3F5F39&\"\"%F3F+F+:F'F+F,F+*&F5F2F8F3F+F+:F'F+F,F+*&F1F2F8F6F+F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "LAP(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF),(\" \"#\"\"\"9%\"\"'*(9$F/F0F/9&F.\"#7F)F):F%F)F*F),$F4F.F)F):F%F)F*F),&*$ F4\"\"$F.*&F3F.F4F/F1F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- \+ Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Depa rtment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYnamed<need<<<====negat<<<nest<neutral <<no <<non<norm<normal <<<notat/ <<<<<<<====note<notic<numb<number<<< numerical<obviou<occur<onlyo(<<<<<<<<<<<<<<<<<<<<<<<====onto<op <<operat<operator <<JD TJX Jx J ȕJ0 앹J J$ 4 |X  x ,   operstor<option<optional7<<<<<<<<<====order<==organiz<orientat<origin << orthogonal<other<otherwis<out <<output<over====own<packag{8< <<<<<<<<<<<<<<<<<<<<<<<<<====page <<paramet#<<<<==== parametero!<<<<<<<<<<<<<<<<<<<<<<<====`}Jl}J| J(}DJ|hJ|JD}J}J|Jx}$J}H doubleint<down<each <<<<<<<element<eliminat<ellips<else<entire <<entr <<eqs<equal <<equat<equation<equival <<eval <<evalf <====evall << <evalm<evaluat <<<====exact<<<<exampl{$<<<<<<<<<<<<<<<<<<<<<<<<<<====J$P`{TEXT JIPJJPTEXTJPJpJP Q {TEXT JJQJKQ0TEXTJ$K QJdK,Q {PARA " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{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:6%%\"xG%\"yG%\"zG6\"6$%)ope ratorG%&arrowGF**(9$\"\"#9%\"\"$9&\"\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Hf:=HESS(f);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #HfG7%7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF,,$*&9%\"\"$9&\"\"% \"\"#F,F,:6%F)F*F+F,F-F,,$*(9$\"\"\"F2F6F4F5\"\"'F,F,:6%F)F*F+F,F-F,,$ *(F;F " 0 "" {MPLTEXT 1 0 10 "Hf(x,y,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%7%,$*&%\"yG\"\"$%\"zG\"\"%\"\"#,$*(%\"xG\"\"\"F'F+ F)F*\"\"',$*(F.F/F'F(F)F(\"\")7%F,,$*(F.F+F'F/F)F*F0,$*(F.F+F'F+F)F(\" #77%F1F7,$*(F.F+F'F(F)F+F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "matrix(Hf(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6# 7%7%,$*&%\"yG\"\"$%\"zG\"\"%\"\"#,$*(%\"xG\"\"\"F*F.F,F-\"\"',$*(F1F2F *F+F,F+\"\")7%F/,$*(F1F.F*F2F,F-F3,$*(F1F.F*F.F,F+\"#77%F4F:,$*(F1F.F* F+F,F.F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=makefunction ([x,y],2*x^2*y+exp(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6$% \"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&9$\"\"#9%\"\"\"F0-%$expG6#F1F2F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Hg:=HESS(g,[a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HgG7$7$:6$%\"aG%\"bG6\"6$%)operat orG%&arrowGF+,$9%\"\"%F+F+:6$F)F*F+F,F+,$9$F1F+F+7$:6$F)F*F+F,F+F4F+F+ :6$F)F*F+F,F+-%$expG6#F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "matrix(Hg(p,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6# 7$7$,$%\"qG\"\"%,$%\"pGF*7$F+-%$expG6#F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Philip B. Yas skin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_ca lc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[hessian]" 2 "linalg[hessian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Dif fops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_M in" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "LAP" 2 "LAP" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "vec _calc[leading_pricipal_minor_determinants]" 2 "vec_calc[leading_pricip al_minor_determinants]" "" }{TEXT -1 2 ". " }}}} ). The functions can always be accessed in the long forms vec_calc[function]$F)F*F+F,F+,$9$F1F+F+7$:6$F)F*F+F,F+F4F+F+ :6$F)F*F+F,F+-%$expG6#F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "matrix(Hg(p,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6# 7$7$,$%\"qG\"\"%,$%\"pGF*7$F+-%$expG6#F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Philip B. Yas skin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_ca lc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[hessian]" <functveccalchesscalculathessianarrownotatcallsequencfcnvarsparameterscalarseveralvariabllistnameusedindependdescriptmatrixsecondpartialderivatactsdefinreturnfunctionchoicoptionaldisplaarrausecommanddifferlinalgpackagexpressrequirspecificatpartcanformonlyafterperformwithalwayaccesslongexamplhfmakefunctexphgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopmultimaxmingradlapleadpricipalminordeterminant@@p@@L@@ &HESSASCBp4HEJ&HESSSE(>\I5:&PH/t1xL1NlC {TEXT " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "M:=[[4, 2, 0, -1], [-3, 5, 1, 0], [0, 2, -3, 1], [2, -4, 1, 3]]; matrix(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7 &7&\"\"%\"\"#\"\"!!\"\"7&!\"$\"\"&\"\"\"F)7&F)F(F,F.7&F(!\"%F.\"\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"%\"\"#\"\"!!\"\"7 &!\"$\"\"&\"\"\"F*7&F*F)F-F/7&F)!\"%F/\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "DM:=LPMD(M);" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leadi ng Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"DG6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#\" #E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"$!#')" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"%!$3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DMG6&\"\"%\"#E!#')!$3$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M2:=submatrix(M,1..2,1..2); det(M2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'MATRIXG6#7$7$\"\"%\"\"#7$!\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M3:=submatrix(M,1..3,1..3); det(M3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'MATRIXG6#7%7%\"\"%\"\"#\"\"!7%!\"$\"\"&\"\"\"7 %F,F+F." }}{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 -1 115 "- Copyright 1995-97 by Arth ur Belmonte and Philip B. Yasskin\n Department of Mathematics, Te xas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "linalg[det]" 2 "linalg[det]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "linalg[submatrix]" 2 "linalg[submatrix]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_Min" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" }{TEXT -1 2 ". " }}}} <<<<<<<<<<<<<<<<<<eXG6#7$7$\"\"%\"\"#7$!\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M3:=submatrix(M,1..3,1..3); det(M3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'MATRIXG6#7%7%\"\"%\"\"#\"\"!7%!\"$\"\"&\"\"\"7 %F,F+F." }}{PARA 11 "" 1 "" {XPPMATH 2short <<sign<simplif <<<<< simplificat <<simplify<sin3<<<<<<<<====sinc<singl<sis <<=siv <<=sol < solut<solution<solv<some <<spac << spacecurv<specif< specificat#<<<<<<<<speed<sph$<<<spher == spherical<spiral ==sqrt<<=squar <<< 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 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 21 " \+ Introduction to the " }{TEXT 256 8 "vec_calc" }{TEXT -1 23 " Pss <<step8< = = = = still<stor<strip<structur<stud << submatric< submatrix<subs<subsequ< substitut<such <<sum<suppos<sure<surfac+E<<<<<<====symbolic <<symmetr<system <<tabl <<tabulat<tang<<< tangential<<<D lengthJO"Gt lexorderJTԤ1  lprintJB~ۮQv  macroJrr~ mapJ-lG8 map2Jd maxJ-lG maxnormJ?? membtrue <<try<type<unassign< unconstrain< unfortunate<unit< universit{<<<<<<<<<<<<<<<<<<<<<<<<<<====up <<upper<upward <<use<<<<<<<usedw?<<<<<<<<<<<<<<<<<<<<<<<<<====uses<using3<<<<<<<<====0 }{CSTYLE "" -1 260 "Courier" 1 10 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 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "MS Sans Serif" 0 1 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 261 16 "vec_calc[JAC] - " }{TEXT -1 104 "Calculates the Jacob ian Matrix of a Coordinate Transformation or of a Parametrized k-Surfa ce in n-Space " }}{PARA 0 "" 0 "" {TEXT 26 20 "Calling Sequences: \n" }{TEXT 256 35 " JAC(T) vec_calc[JAC](T)" }}{PARA 0 "" 0 "" {TEXT 258 40 " JAC(T,vars) vec_calc[JAC](T,vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 12 " T - " }{TEXT -1 104 "a coordinate transformation in the form of a vector or list of n arrow-defined functions of k var iables " }}{PARA 0 "" 0 "" {TEXT 259 12 " vars - " }{TEXT -1 80 "a list of k names to be used as the independent variables of the transf ormation " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 58 "The Jacobian matrix of \+ a coordinate transformation is the " }{TEXT 260 5 "n x k" }{TEXT -1 118 " matrix whose ij'th entry is the partial derivatives of the i'th \+ component function with respect to the j'th variable." }}{PARA 15 "" 0 "" {TEXT -1 179 "JAC returns a list of lists of the first partial de rivatives of the coordinate transformation. The choice of variables i s optional. To display the Jacobian as an array, use the " } {HYPERLNK 17 "matrix" 2 "matrix" "" }{TEXT -1 9 " command." }}{PARA 15 "" 0 "" {TEXT -1 29 "JAC differs from the command " }{HYPERLNK 17 " jacobian" 2 "jacobian" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg " 2 "linalg" "" }{TEXT -1 152 " package:\nlinalg[jacobian] acts on a v ector or list of expressions and returns a matrix of expressions. It \+ requires the specification of the variables." }}{PARA 15 "" 0 "" {TEXT -1 226 "The function JAC is part of the vec_calc package, and so can be used in the form JAC only after performing the command with(ve c_calc) or with(vec_calc, JAC). The function can always be accessed i n the long form vec_calc[JAC]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T:=make function([u,v],[u^2+v^2,u+v,u*v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"TG7%:6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF*,&*$9$\"\"#\"\"\"*$9%F1 F2F*F*:F'F*F+F*,&F0F2F4F2F*F*:F'F*F+F**&F0F2F4F2F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "JT:=JAC(T); matrix(JT(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#JTG7%7$:6$%\"uG%\"vG6\"6$%)operatorG%&arrowG F+,$9$\"\"#F+F+:6$F)F*F+F,F+,$9%F1F+F+7$\"\"\"F77$:6$F)F*F+F,F+F5F+F+: 6$F)F*F+F,F+F0F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7$ ,$%\"uG\"\"#,$%\"vGF*7$\"\"\"F.7$F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "T:=makefunction([rho,theta,phi], [rho*sin(phi)*cos(th eta),\nrho*sin(phi)*sin(theta),rho*cos(phi)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG7%:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowG F+*(9$\"\"\"-%$sinG6#9&F1-%$cosG6#9%F1F+F+:F'F+F,F+*(F0F1F2F1-F3F8F1F+ F+:F'F+F,F+*&F0F1-F7F4F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "JT:=JAC(T,[r,t,p]); matrix(JT(rho,theta,phi));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#JTG7%7%:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&a rrowGF,*&-%$sinG6#9&\"\"\"-%$cosG6#9%F5F,F,:6%F)F*F+F,F-F,,$*(9$F5F1F5 -F2F8F5!\"\"F,F,:6%F)F*F+F,F-F,*(F>F5-F7F3F5F6F5F,F,7%:6%F)F*F+F,F-F,* &F1F5F?F5F,F,:6%F)F*F+F,F-F,*(F>F5F1F5F6F5F,F,:6%F)F*F+F,F-F,*(F>F5FDF 5F?F5F,F,7%:6%F)F*F+F,F-F,FDF,F,\"\"!:6%F)F*F+F,F-F,,$*&F>F5F1F5F@F,F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%*&-%$sinG6#%$phiG \"\"\"-%$cosG6#%&thetaGF-,$*(%$rhoGF-F)F--F*F0F-!\"\"*(F4F--F/F+F-F.F- 7%*&F)F-F5F-*(F4F-F)F-F.F-*(F4F-F8F-F5F-7%F8\"\"!,$*&F4F-F)F-F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JAC_DET(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF(, $*&-%$sinG6#9&\"\"\"9$\"\"#!\"\"F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Philip B. Yas skin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_ca lc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[jacobian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Di ffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JAC_DET" 2 "JAC_DET" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" } {TEXT -1 2 ". " }}}} @O~ \+} 25  F+F-F.F- 7%*&F)F-F5F-*(F4F-F)F-F.F-*(F4F-F8F-F5F-7%F8\"\"!,$*&F4F-F)F-F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JAC_DET(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF(, $*&-%$sinG6#9&\"\"\"9$\"\"#!\"\hessian <<<<hf <<hg< horizontal <<howev <<identif<ii<<<iii<<<ij<includ<<<< incorrect<independ/ <<<<<<<<<<<indicat<====inert<input <<insid<instead <<ints<<<====integral$<<==== integrand<integrat<====interior< intermediat<==== interpret<into<<< introduct<--..//00112233445566778899::;;<<=involv <<iv <<jac(<<<< <jacobian <<<jame<jared<jerk<<<jt <<just<ken<lagrang<lambda<lap<<<< laplacian <<lead&<<<<<<left <<len<<< length<<<lett<letter<libnam<LPTPXP\P`PdPhPlPP|||||||||||} }0}(}P}H}p}h}}}}}}}}}~}~~ <4XPphģأ4,TLphĤ " 0 "" {MPLTEXT 1 0 15 "with(vec _calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "T:=makefunction( [rho,theta,phi], [rho*sin(phi)*cos(theta),\nrho*sin(phi)*sin(theta),rh o*cos(phi)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG7%:6%%$rhoG%&th etaG%$phiG6\"6$%)operatorG%&arrowGF+*(9$\"\"\"-%$sinG6#9&F1-%$cosG6#9% F1F+F+:F'F+F,F+*(F0F1F2F1-F3F8F1F+F+:F'F+F,F+*&F0F1-F7F4F1F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JT:=JAC(T);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#JTG7%7%:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG% &arrowGF,*&-%$sinG6#9&\"\"\"-%$cosG6#9%F5F,F,:6%F)F*F+F,F-F,,$*(9$F5F1 F5-F2F8F5!\"\"F,F,:6%F)F*F+F,F-F,*(F>F5-F7F3F5F6F5F,F,7%:6%F)F*F+F,F-F ,*&F1F5F?F5F,F,:6%F)F*F+F,F-F,*(F>F5F1F5F6F5F,F,:6%F)F*F+F,F-F,*(F>F5F DF5F?F5F,F,7%:6%F)F*F+F,F-F,FDF,F,\"\"!:6%F)F*F+F,F-F,,$*&F>F5F1F5F@F, F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JAC_DET(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arr owGF(,$*&-%$sinG6#9&\"\"\"9$\"\"#!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "det(JT(rho,theta,phi)); simplify(\");" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,**(-%$sinG6#%$phiG\"\"$-%$cosG6#%&thetaG\"\"#%$ rhoGF.!\"\"*(F%F)-F&F,F.F/F.F0**F/F.F%\"\"\"F2F.-F+F'F.F0**F/F.F5F.F*F .F%F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#%$phiG\"\"\"%$ rhoG\"\"#!\"\"" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1 995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of M athematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See A lso:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JAC" 2 "JAC" "" }{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 2 ", " }{HYPERLNK 17 " Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion2D " 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConver sion3D" 2 "CoordConversion3D" "" }{TEXT -1 2 ". " }}}} 486%/$MATH 20 "6#,**(-%$sinG6#%$phiG\"\"$-%$cosG6#%&thetaG\"\"#%$ rhoGF.!\"\"*(F%F)-F&F,F.F/F.F0**F/F.F%\"\"\"F2F.-F+F'F.F0**F/F.F5F.F*F .F%F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#%$phiG\"\"\"%$ rhoG\"\"#!\"\"" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1 995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of M athematics, Texas A&M University " }}{< {VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 " " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "g:=makefunction([x,y,z],x^2+exp(y)*sin(z)); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF*,&*$9$\"\"#\"\"\"*&-%$expG6#9%F2-%$sinG6#9&F2F2F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G:=GRAD(g); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG F+,$9$\"\"#F+F+:F'F+F,F+*&-%$expG6#9%\"\"\"-%$sinG6#9&F8F+F+:F'F+F,F+* &F4F8-%$cosGF;F8F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "POT (G,'h'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(h); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF(,&*$9$\"\"#\"\"\"*&-%$e xpG6#9%F0-%$sinG6#9&F0F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "POT(G,'h',[a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6%%\"aG%\"bG%\"cG6\"6$%)operatorG%&arrowGF(,&*$9$\" \"#\"\"\"*&-%$expG6#9%F0-%$sinG6#9&F0F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "F:=makefunction([x,y,z],[2*y, exp(y)*sin(z), exp(y )*cos(z)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%:6%%\"xG%\"yG% \"zG6\"6$%)operatorG%&arrowGF+,$9%\"\"#F+F+:F'F+F,F+*&-%$expG6#F0\"\" \"-%$sinG6#9&F7F+F+:F'F+F,F+*&F4F7-%$cosGF:F7F+F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "POT(F,'f'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"fG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "linalg[potential]" 2 "linalg[potential] " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "VEC_PO T" 2 "VEC_POT" "" }{TEXT -1 2 ". " }}}} %$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF(, $*&-%$sinG6#9&\"\"\"9$\"\"#!\zG6\"6$%)operatorG%&arrowGF+,$9%\"\"#F+F+:F'F+F,F+*&-%$expG6#F0\"\" \"-%$sinG6#9&F7F+F+:F'F+F,F+*&F4F7-%$cosGF:F7F+F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "POT(F,'f'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"fG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_caparametr << parametric<<== parametriz< parametrizat< parenthes<park<partk<<<<<<<<<<<<<<<<<<<<<<====partial<<<path<performs)<<<<<<<<<<<<<<<<<<<<<<<<====period<permit< perpendicular<phi<<<==philip{<<<<<<<<<<<<<<<<<<<<<<<<<<====A 15 "" 0 "" {HYPERLNK 17 "Curve" 2 "Curve" "" }}{PARA 270 "" 0 "" {TEXT -1 9 " frenet" }} {PARA 294 "" 0 "" {TEXT -1 36 " curve_velocity Cv" } }{PARA 271 "" 0 "" {TEXT -1 36 " curve_acceleration Ca" }}{PARA 272 "" 0 "" {TEXT -1 36 " curve_jerk Cj " }}{PARA 273 ""pi<<<====plot <<<plott<point<<<<<<polar<<<< polynomial<posit<<<<possib<====possibl<pot/<<<<<<< potential<< < precedenc <<pres<previou<pricipal< principal&<<<<<< \J8 J`J sinJG`J,J thetaJJ`PJ L,tJ phiJJ<functveccalcjacdetcalculatjacobiandeterminantcoordinattransformatcallsequencvarsparameterformvectorlistarrowdefinfunctionvariablnameusedindependdescriptmatrixreturnchoicoptionalpartpackagcanonlyafterperformcommandwithalwayaccesslongexamplmakefunctrhothetaphisincosjtsimplifcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsolinalgdiffopmuintcoordconversyasskindepartmmathematictexauniversitalsodiffopcurl" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 261 20 "vec_calc[VEC_POT] - " }{TEXT -1 81 "Calculates the Ve ctor Potential of a Vector Field in Arrow Notation If It Exists " }} {PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 257 51 " VEC_POT(F,'A') vec_calc[VEC_ POT](F,'A')" }}{PARA 0 "" 0 "" {TEXT 259 56 " VEC_POT(F,'A',vars) \+ vec_calc[VEC_POT](F,'A',vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parame ters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 12 " F - " } {TEXT -1 81 "a vector field in the form of a list of 3 arrow-defined f unctions of 3 variables " }}{PARA 0 "" 0 "" {TEXT 258 12 " 'A' - \+ " }{TEXT -1 49 "the name for the vector potential to be returned " }} {PARA 0 "" 0 "" {TEXT 260 12 " vars - " }{TEXT -1 57 "a list of 3 \+ names to be used as the independent variables" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 133 "VEC_POT determines whether a given vector field is the c url of a vector potential, and determines that vector potential if it \+ exists." }}{PARA 15 "" 0 "" {TEXT -1 92 "VEC_POT returns true if the v ector field F has a vector potential, and false if it does not." }} {PARA 15 "" 0 "" {TEXT -1 150 "If a vector potential for F exists, it \+ will be assigned to the name given in the second argument A. This sec ond argument must be contained in quotes." }}{PARA 15 "" 0 "" {TEXT -1 33 "VEC_POT differs from the command " }{HYPERLNK 17 "vecpotent" 2 "vecpotent" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg" 2 "linalg " "" }{TEXT -1 171 " package:\nlinalg[vecpotent] acts on a vector or l ist of expressions and returns the potential as a vector of expression s. It requires the specification of the variables. " }}{PARA 15 "" 0 "" {TEXT -1 242 "The function VEC_POT is part of the vec_calc package, and so can be used in the form VEC_POT only after performing the comm and with(vec_calc) or with(vec_calc, VEC_POT). The function can alway s be accessed in the long form vec_calc[VEC_POT]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A:=makefunction([x,y,z],[x+y+z, x*y*z, x*y+y*z+z*x]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%:6%%\"xG%\"yG%\"zG6\"6$%)o peratorG%&arrowGF+,(9$\"\"\"9%F19&F1F+F+:F'F+F,F+*(F0F1F2F1F3F1F+F+:F' F+F,F+,(*&F0F1F2F1F1*&F2F1F3F1F1*&F3F1F0F1F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C:=CURL(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"CG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(9$\"\"\"9&F1*& F0F19%F1!\"\"F+F+:F'F+F,F+,(F1F1F4F5F2F5F+F+:F'F+F,F+,&*&F4F1F2F1F1F5F 1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VEC_POT(C,'B');" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6%%\"xG %\"yG%\"zG6\"6$%)operatorG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*$F.\"\"# #F2F4F1F/F)F):F%F)F*F),(*&F.F/9$F/F2F3F5*(F9F/F1F/F.F/F/F)F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "VEC_POT(C,'B',[a,b,c]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%:6%%\"aG %\"bG%\"cG6\"6$%)operatorG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*$F.\"\"# #F2F4F1F/F)F):F%F)F*F),(*&F.F/9$F/F2F3F5*(F9F/F1F/F.F/F/F)F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VEC_POT(A,'V');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"VG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[vecpotent]" 2 "linalg[vecpotent]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DIV" 2 "DIV" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 2 ". " }}}} }|G(4 0orJT4 11 "" 1 "" {XPPMATH 20 "6#7%:6%%\"aG %\"bG%\"cG6\"6$%)operatorG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*$F.\"\"# #F2F4F1F/F)F):F%F)F*F),(*&F.F/9$F/F2F3F5*(F9F/F1F/F.F/F/F)F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VEC_POT(A,'V');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"VG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte a nd Philip B. Yasskin\n Department of Mathematics, Texas A&M Unive rsity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[vecpotent]" 2 "linalg[vecpotent]" "" }{TEXT =%{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 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 }{CSTYLE " " -1 263 "Courier" 1 10 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 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 1<{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 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 }{CSTYLE " " -1 263 "Courier" 1 10 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 }{CSTYLE "" -1 265 "Courier" 1 10 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 }{CSTYLE "" -1 267 "Courier" 1 10 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fix ed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }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 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 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 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 27 " vec_calc[Multipleint] - " }{TEXT -1 37 "Display s an Inert Mulitiple Integral " }}{PARA 0 "" 0 "" {TEXT 257 27 " vec _calc[multipleint] - " }{TEXT -1 67 "Computes a Multiple Integral poss ibly Displaying Intermediate Steps" }}{PARA 0 "" 0 "" {TEXT 26 8 "Alia ses:" }{TEXT -1 50 " - The aliases can be used after execution of the \+ " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 9 " command. " }}{PARA 258 "" 0 "" {TEXT -1 13 " Muint = " }{TEXT 259 11 "Multi pleint" }}{PARA 259 "" 0 "" {TEXT -1 13 " muint = " }{TEXT 260 11 "multipleint" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{TEXT 262 11 "Mul tipleint" }{TEXT -1 55 "(F,x1,x2,...,xn) Muint(...) \+ vec_calc[" }{TEXT 263 11 "Multipleint" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 261 11 "multipleint" }{TEXT -1 55 "(F,x1,x2,...,xn) muint(...) vec_calc[" }{TEXT 264 11 "multipleint" }{TEXT -1 6 "](...)" }}{PARA 260 "" 0 "" {TEXT -1 3 " " }{TEXT 265 11 "multipleint" }{TEXT -1 55 "(F,x1,x2,...,xn,`step`) muint(...,`step`) vec_calc[" }{TEXT 266 11 "multipleint" }{TEXT -1 13 "](...,`step`)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }} {PARA 0 "" 0 "" {TEXT 258 15 " F - " }{TEXT -1 41 "an expres sion to be used as an integrand " }}{PARA 0 "" 0 "" {TEXT 267 15 " x 1 ... xn - " }{TEXT -1 112 "each is a name or a name = range, to speci fy a variable of integration and optionally its limits of integration. " }}{PARA 0 "" 0 "" {TEXT 268 15 " `step` - " }{TEXT -1 82 "an op tional parameter to indicate that the intermediate steps should be dis played." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 298 "MuInt displays a multiple in tegral, where the first argument is the integrand and the following ar guments are the variables of integration, which can include numerical \+ ranges. The variables appear in the order they are to be evaluated. \+ The integral can then be evaluated using the value command. " }}{PARA 15 "" 0 "" {TEXT -1 76 "muint calculates a multiple integral without f irst displaying the integral. " }}{PARA 15 "" 0 "" {TEXT -1 220 "muint with the `step` parameter displays the integral and then calculates i t while displaying all the intermediate steps. You do not need the ba ckquotes around `step` if the variable step has not been assigned a va lue. " }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part of th e vec_calc package, and so can be used by name only after performing t he command with(vec_calc) or with(vec_calc,function). The functions c an always be accessed in the long forms vec_calc[function]. The alias es can be used only after performing the command " }{HYPERLNK 17 "vc_a liases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Muint(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\"\"$%\"zG\"\"#/F+;\"\"\"F0/F-;F.F,/F/;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "muint(x^4*y^3*z^2,x=1..2,y=3..4,z=5..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "muint(x^4*y^3*z^2,x=1..2,y=3..4,z=5..6,`step`):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*(%\"xG\"\"%%\"yG\" \"$%\"zG\"\"#/F+;\"\"\"F0/F-;F.F,/F/;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$-F&6$*&-%'VECTORG6#7#,$*(%\"xG\"\"&%\"yG \"\"$%\"zG\"\"##\"\"\"F2F8-%'MATRIXG6#7%7#&%!G6#/F1F67#F?7#&F?6#/F1F8F 8/F3;F4\"\"%/F5;F2\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$In tG6$-F&6$,$*&%\"yG\"\"$%\"zG\"\"##\"#J\"\"&/F,;F-\"\"%/F.;F2\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$*&-%'VECTORG6#7#,$*&%\" yG\"\"%%\"zG\"\"##\"#J\"#?\"\"\"-%'MATRIXG6#7%7#&%!G6#/F/F07#F=7#&F=6# /F/\"\"$F6/F1;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$I ntG6$,$*$%\"zG\"\"##\"%&3\"\"\"%/F*;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#,$*$%\"zG\"\"$#\"%&3\"\"#7\"\"\"- %'MATRIXG6#7%7#&%!G6#/F,\"\"'7#F87#&F86#/F,\"\"&F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"~G#\"&N()*\"#7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 1 "," }{HYPERLNK 17 "Doubleint" 2 "Doubleint" "" }{TEXT -1 1 " " }{HYPERLNK 17 "Tripleint" 2 "Tripleint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JAC_DET" 2 "JAC_DET" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Line_int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vect or" "" }{TEXT -1 2 ". " }}}} *\"#7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 1 "," }{HYPERLNK 17 "Doubleint" 2 "Doubleint" "" }{TEXT -1 1 " " }{HYvalu3'<<<<<<<<====var====variablOK<<<<<<<<<<<<<<<====vars+(< <<<<<<<<<vcC+<<<<<<<<<<<<====vec{I<<< <<< < << < <<< << < < < < < << < < << = = = = vecpot<vectorS< <<<<<<<<<<<<<< <====velocit<<<<vers<vertical <<warren<we < well<were <<wheth <<whil<====whos<will<<<with{n<<<<<<<<<<<<<<<<<<<<<<<<<<====without<====won<work <<written<xn<yasskin{#<<<<<<<<<<<<<<<<<<<<<<<<<<====you' <<<<<====zero<Th{HYPERLNK pHYPERLNKx{TEXT uF([x,y,z],[x^2+y^3+x*y*z^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {Xassign<<<====at <<< automatical<availabl<avoid <<axes<axis <<back <<backquot<====basical<becom<beginn<belmont{ <<<<<<<<<<<<<<<<<<<<<<<<<<====below<<<binormal<<<throughout the text \"Vector Calculus with Maple V\" by Arthur Belmonte and Philip B. Yasskin, to be published, \+ 1997." }}{PARA 15 "" 0 "" {TEXT -1 289 "The vec_calc commands were wri tten by A. Belmonte and P. Yasskin. The commands were organized into \+ a package by James Warren and P. Yasskin. The help pages were first w ritten by David Arnold, J. Warren and P. Yasskin and converted to Rele ase 4 by Ken Parker, Jared Teslow and P. Yasskin." }}{PARA 15 "" 0 "" {TEXT -1 135 "@ Copyright 1995-97 by Art<functveccalcpotcalculatvectorpotentialfieldarrownotatexistcallsequencvarsparameterformlistdefinfunctionvariablnamereturnusedindependdescriptdeterminwhethgivencurltruefalsdoeswillassignsecondargummustcontainquotdiffercommandvecpotlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctevalcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopdivcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsointdoubleinttripleintjacdetlinescalarvectorsurfac<<<<<theta(<<<<<they<think<third< throughout<time<top<topic<tors<<< transformat <<<<troubl<Jۦbl IPiJbYl IRootOfJt3$m 10 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 }{CSTYLE "" -1 268 "Courier" 1 10 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 } {CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }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 }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 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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 31 " vec_calc[Line_int_scalar] - " }{TEXT -1 43 "Dis plays a Line Integral of a Scalar Field " }}{PARA 0 "" 0 "" {TEXT 257 31 " vec_calc[line_int_scalar] - " }{TEXT -1 81 "Computes a Line Int egral of a Scalar Field possibly Displaying Intermediate Steps" }} {PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliases can be used after execution of the " }{HYPERLNK 17 "vc_aliases" 2 "vc_ali ases" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 11 " L is = " }{TEXT 261 15 "Line_int_scalar" }}{PARA 259 "" 0 "" {TEXT -1 11 " lis = " }{TEXT 262 15 "line_int_scalar" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{TEXT 265 15 "Line_int_scalar" }{TEXT -1 33 "(fcn,r, var=rng) Lis(...)" }}{PARA 257 "" 0 "" {TEXT -1 17 " v ec_calc[" }{TEXT 267 15 "Line_int_scalar" }{TEXT -1 6 "](...)" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 263 15 "line_int_scalar" } {TEXT -1 33 "(fcn,r,var=rng) lis(...)" }}{PARA 256 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 268 15 "line_int_scalar" } {TEXT -1 6 "](...)" }}{PARA 260 "" 0 "" {TEXT -1 3 " " }{TEXT 264 15 "line_int_scalar" }{TEXT -1 40 "(fcn,r,var=rng,`step`) lis(...,`s tep`)" }}{PARA 260 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 269 15 "line_int_scalar" }{TEXT -1 13 "](...,`step`)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 258 14 " fcn \+ - " }{TEXT -1 51 "a scalar function of n variables in arrow notation \+ " }}{PARA 0 "" 0 "" {TEXT 259 14 " r - " }{TEXT -1 75 "a curv e in the form of a list of n arrow-defined functions of one parameter " }}{PARA 0 "" 0 "" {TEXT 266 14 " var - " }{TEXT -1 47 "the va riable of integration and curve parameter" }}{PARA 0 "" 0 "" {TEXT 270 14 " rng - " }{TEXT -1 44 "the range of the parameter to in tegrate over" }}{PARA 0 "" 0 "" {TEXT 260 14 " `step` - " }{TEXT -1 82 "an optional parameter to indicate that the intermediate steps s hould be displayed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descripti on:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 276 "Line_int_scalar \+ displays the line integral of a scalar field, where the first argument is the scalar field, the second argument is the curve and the third a rgument is the variable and range of integration. The integral can th en be evaluated using the value or evalf command. " }}{PARA 15 "" 0 " " {TEXT -1 102 "line_int_scalar calculates the line integral of a scal ar field without first displaying the integral. " }}{PARA 15 "" 0 "" {TEXT -1 253 "line_int_scalar with the `step` parameter displays the l ine integral of a scalar field and then calculates it while displaying all the intermediate steps. You do not need the backquotes around `s tep` if the variable step has not been assigned a value. " }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part of the vec_calc package , and so can be used by name only after performing the command with(ve c_calc) or with(vec_calc,function). The functions can always be acces sed in the long forms vec_calc[function]. The aliases can be used onl y after performing the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_alia ses" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Exampl es:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with( vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f :=makefunction([x,y,z],2/9*y*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,$*&9%\"\"\"9&F1#\"\"# \"\"*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r:=makefunctio n(t,[2*t,3*sin(t),3*cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"r G7%:6#%\"tG6\"6$%)operatorG%&arrowGF),$9$\"\"#F)F):F'F)F*F),$-%$sinG6# F.\"\"$F)F):F'F)F*F),$-%$cosGF4F5F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Line_int_scalar(f,r,t=0..Pi/2); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(-%$sinG6#%\"tG\"\"\"-%$cosGF*F, \"#8#F,\"\"#F1/F+;\"\"!,$%#PiGF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$ \"#8#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "line_in t_scalar(f,r,t=0..Pi/2,`step`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ IntG6$,$*(-%$sinG6#%\"tG\"\"\"-%$cosGF*F,\"#8#F,\"\"#F1/F+;\"\"!,$%#Pi GF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#*&-%$sinG 6#%\"tG\"\"#\"#8#\"\"\"F/F2-%'MATRIXG6#7%7#&%!G6#/F.,$%#PiGF17#F97#&F9 6#/F.\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G,&*&-%$sinG6#,$% #PiG#\"\"\"\"\"#F.\"#8F,F-*&-F(6#\"\"!F.F/F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#8#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=MF([x,y],3*x*sin(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\" \"-%$sinG6#9%F0\"\"$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R:=MF(t,[ln(t),2*(-t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$% #lnG:6#%\"tG6\"6$%)operatorG%&arrowGF*,$9$!\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Line_int_scalar(g,R,t=1..2); value(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$**-%#lnG6#%\"tG\"\"\"-%$sin G6#,$F+\"\"#F,F+!\"\",&*$F+F1\"\"%F,F,#F,F1!\"$/F+;F,F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$intG6$,$**-%#lnG6#%\"tG\"\"\"-%$sinG6#,$F+\" \"#F,F+!\"\",&*$F+F1\"\"%F,F,#F,F1!\"$/F+;F,F1" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+#[:[K$!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h:=MF([x,y, z,w],x*z*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG:6&%\"xG%\"yG%\" zG%\"wG6\"6$%)operatorG%&arrowGF+*(9$\"\"\"9&F19'F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:=MF(t,[2*t,t^2,t^2,2*t^3/3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7&:6#%\"tG6\"6$%)operatorG%&arro wGF),$9$\"\"#F)F):F'F)F*F)*$F.F/F)F):F'F)F*F)F1F)F):F'F)F*F),$*$F.\"\" $#F/F6F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "line_int_scal ar(h,r,t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$G\"\"$*=" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "line_int_scalar(h,r,t=0..1,' step'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&%\"tG\"\"',&\" \"#\"\"\"*$F(F+F+F,#\"\"%\"\"$/F(;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#,&*$%\"tG\"\"*#\"\")\"#F*$F,\"\"( #F/\"#@\"\"\"-%'MATRIXG6#7%7#&%!G6#/F,F57#F<7#&F<6#/F,\"\"!F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"$G\"\"$*=" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makef unction" "" }{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 "int" 2 " int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_v ector" "" }{TEXT -1 2 ". " }}}}  VECTORJEͬRJ,$SJ,dS." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"$G\"\"$*=" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Phi lip B. Yasskin\n Department of Mathematics, Texas A&M University \+ " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makef unction" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2able<abov <<absolut << accelerat<<< <accessg<<<<<<<<<<<<<<<<<<<<<====acknowledgement<action<acts# <<<<<<<<advis<after{5<<<<<<<<<<<<<<<<<<<<<<<<<<====algebra << algebraic<alia<<<<<<TEXT 26 1 "." }}}} @˸ .'"XT 1 0 11 "rad2deg(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG\"\"\"%#Pi G!\"\"\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2rad(45); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{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 "deg2 rad(45.);"aliasCN<<<<<<<<<<<<====all+<<<<<<====allvalu<also{ <<<<<<<<<<<<<<<<<<<<<<<<<<====alternat<alwayg<<<<<<<<<<<<<<<<<<<<<====am<analys<analysi <<angl <<<answ< "f:=MF(t,[t,t^2,t^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%:6#%\"tG6\"6$%)operatorG%&arrowGF)9$F)F):F'F)F* F)*$F-\"\"#F)F):F'F)F*F)*$F-\"\"$F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 56 "A scalar function of several variables (e.g. a density):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f:=makefunction([x,y,z,t],x^2*sin(t *y)+ln(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6&%\"xG%\"yG%\"z G%\"tG6\"6$%)operatorG%&arrowGF+,&*&9$\"\"#-%$sinG6#*&9'\"\"\"9%F8F8F8 -%#lnG6#9&F=${VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 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 }{CSTYLE " " -1 263 "Courier" 1 10 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 }{CSTYLE "" -1 265 "Courier" 1 10 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 }{CSTYLE "" -1 267 "Courier" 1 10 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 }{CSTYLE "" -1 269 "Courier" 1 10 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 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }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 }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 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 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 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 31 " vec_calc[Line_int_vector] - " }{TEXT -1 43 "Dis plays a Line Integral of a Vector Field " }}{PARA 0 "" 0 "" {TEXT 257 31 " vec_calc[line_int_vector] - " }{TEXT -1 81 "Computes a Line Int egral of a Vector Field possibly Displaying Intermediate Steps" }} {PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliases can be used after execution of the " }{HYPERLNK 17 "vc_aliases" 2 "vc_ali ases" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 11 " L iv = " }{TEXT 261 15 "Line_int_vector" }}{PARA 259 "" 0 "" {TEXT -1 11 " liv = " }{TEXT 262 15 "line_int_vector" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{TEXT 265 15 "Line_int_vector" }{TEXT -1 31 "(F,r,va r=rng) Liv(...)" }}{PARA 257 "" 0 "" {TEXT -1 17 " vec _calc[" }{TEXT 267 15 "Line_int_vector" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 263 15 "line_int_vector" }{TEXT -1 31 "(F,r,var=rng) liv(...)" }}{PARA 256 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 268 15 "line_int_vector" }{TEXT -1 6 "]( ...)" }}{PARA 260 "" 0 "" {TEXT -1 3 " " }{TEXT 264 15 "line_int_vec tor" }{TEXT -1 38 "(F,r,var=rng,`step`) liv(...,`step`)" }}{PARA 260 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 269 15 "line_int_ve ctor" }{TEXT -1 13 "](...,`step`)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Para meters: " }}{PARA 0 "" 0 "" {TEXT 258 14 " F - " }{TEXT -1 85 "a vector field in the form of a list of n functions of n variables in arrow notation " }}{PARA 0 "" 0 "" {TEXT 259 14 " r  - " } {TEXT -1 75 "a curve in the form of a list of n arrow-defined function s of one parameter" }}{PARA 0 "" 0 "" {TEXT 270 14 " var - " } {TEXT -1 47 "the variable of integration and curve parameter" }}{PARA 0 "" 0 "" {TEXT 266 14 " rng - " }{TEXT -1 44 "the range of the parameter to integrate over" }}{PARA 0 "" 0 "" {TEXT 260 14 " `step ` - " }{TEXT -1 82 "an optional parameter to indicate that the inter mediate steps should be displayed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 276 " Line_int_vector displays the line integral of a vector field, where th e first argument is the vector field, the second argument is the curve and the third argument is the variable and range of integration. The integral can then be evaluated using the value or evalf command. " }} {PARA 15 "" 0 "" {TEXT -1 102 "line_int_vector calculates the line int egral of a vector field without first displaying the integral. " }} {PARA 15 "" 0 "" {TEXT -1 253 "line_int_vector with the `step` paramet er displays the line integral of a vector field and then calculates it while displaying all the intermediate steps. You do not need the bac kquotes around `step` if the variable step has not been assigned a val ue. " }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part of the vec_calc package, and so can be used by name only after performing th e command with(vec_calc) or with(vec_calc,function). The functions ca n always be accessed in the long forms vec_calc[function]. The aliase s can be used only after performing the command " }{HYPERLNK 17 "vc_al iases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}}{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%:6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF+,$*&9$\"\"#9%\"\"\"\"$V\"F+F+:F'F+F,F+,$*&F3F 49&F4$!$:(!\"\"F+F+:F'F+F,F+,$*&F1F4F9F4$\"#UF " 0 "" {MPLTEXT 1 0 37 "r:=makefunction(t,[2*t^3,3*t^4,t^2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%:6#%\"tG6\"6$%)operatorG%&arro wGF),$*$9$\"\"$\"\"#F)F):F'F)F*F),$*$F/\"\"%F0F)F):F'F)F*F)*$F/F1F)F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Line_int_vector(F,r,t=0 ..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$%\"tG\"#7$\"&'H5 \"\"!*$F(\"\"*$!%uDF,*$F(\"\"'$\"++++!o\"!\")/F(;F,\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "line_int_vector(F,r,t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$P&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "G:=MF([x,y],[x^3,x^5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF**$9$\"\" $F*F*:F'F*F+F**$F/\"\"&F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "R:=MF(t,[(cos(t))^3,(sin(t))^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$:6#%\"tG6\"6$%)operatorG%&arrowGF)*$-%$cosG6#9$\"\"$F)F):F' F)F*F)*$-%$sinGF0F2F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " Liv(G,R,t=0..Pi/2); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$,(*&-%$cosG6#%\"tG\"#6-%$sinGF*\"\"\"!\"$*$F(\"#;\"\"$*$F(\"#=F0/ F+;\"\"!,$%#PiG#F/\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#PiG#\"% X@\"'s58#!\"\"\"\"%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "H:=MF([x,y,z,w],[x^2,x*w,w,z^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7&:6&%\"xG%\"yG%\"zG%\"wG6\"6$%)operatorG%&arrowGF,*$9$\"\"#F ,F,:F'F,F-F,*&F1\"\"\"9'F5F,F,:F'F,F-F,F6F,F,:F'F,F-F,*$9&F2F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r:=MF(t,[sin(t),cos(t),sin(t ),cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7&%$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$,(*$-%$cosG6#%\"tG\" \"#\"\"\"-%$sinGF*!\"\"*&F.F-F(F,F-/F+;\"\"!,$%#PiG#F-F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#,**&-%$sinG6#%\"tG\"\"\"-% $cosGF.F0#F0\"\"#F/F3F1F0*$F1\"\"$#!\"\"F6F0-%'MATRIXG6#7%7#&%!G6#/F/, $%#PiGF37#F?7#&F?6#/F/\"\"!F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"~ G,0*&-%$sinG6#,$%#PiG#\"\"\"\"\"#F--%$cosGF)F-F,F+#F-\"\"%F/F-*$F/\"\" $#!\"\"F4*&-F(6#\"\"!F--F0F9F-#F6F.F;F6*$F;F4#F-F4" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&#!\"#\"\"$\"\"\"%#PiG#F'\"\"%" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "- Copyright 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " } }{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_calc" 2 " vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makefunc tion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 " int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_v ector" "" }{TEXT -1 2 ". " }}}} J,/J-$/h0J8-,/ARA 11 "" 1 " " {XPPMATH 20 "6#,&#!\"#\"\"$\"\"\"%#PiG#F'\"\"%" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "- Copyright 1995-97 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 "vec_calc" 2 " vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makefunc tion" "" }{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 "int" 2 " int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" " " }{TEXT -1 2 ",=1functionveccalclineintvectordisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandlivcallsequencvarrngparameterformlistvariablarrownotatcurvdefinparametintegratrangoveroptionalindicatdescriptfirstargumsecondthirdevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignthespartpackagnameonlyperformfunctalwayaccesslongexamplmakefunctmfcossinpicopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintscalarsurfac{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }} {TEXT 256 15 "Line_int_vector" } } L*timeJ=nKh L*kernel/transposeJh L*traperrorJ i L*trunc L*typeJoL`i L*typematchJ̐i L*unamesJh(vi L*unio=?&{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 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 }{CSTYLE "" -1 261 "Courier" 1 10 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 }{CSTYLE " " -1 263 "Courier" 1 10 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 }{CSTYLE "" -1 265 "Courier" 1 10 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 }{CSTYLE "" -1 267 "Courier" 1 10 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 }{CSTYLE "" -1 269 "Courier" 1 10 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 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }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 }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 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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 34 " vec_calc[Surface_int_scalar] - " }{TEXT -1 46 " Displays a Surface Integral of a Scalar Field " }}{PARA 0 "" 0 "" {TEXT 257 34 " vec_calc[surface_int_scalar] - " }{TEXT -1 84 "Comput es a Surface Integral of a Scalar Field possibly Displaying Intermedia te Steps" }}{PARA 0 "" 0 "" {TEXT 26 8 "Aliases:" }{TEXT -1 50 " - The aliases can be used after execution of the " }{HYPERLNK 17 "vc_aliase s" 2 "vc_aliases" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 11 " Sis = " }{TEXT 261 18 "Surface_int_scalar" }}{PARA 259 "" 0 "" {TEXT -1 11 " sis = " }{TEXT 264 18 "surface_int_scala r" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{TEXT 262 18 "Surface_int_scalar " }{TEXT -1 45 "(fcn,R,var1=rng1,var2=rng2) Sis(...)" }} {PARA 257 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 263 18 "Surfa ce_int_scalar" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 265 18 "surface_int_scalar" }{TEXT -1 45 "(fcn,R,var1=rng1,v ar2=rng2) sis(...)" }}{PARA 256 "" 0 "" {TEXT -1 17 " \+ vec_calc[" }{TEXT 267 18 "surface_int_scalar" }{TEXT -1 6 "](...)" }} {PARA 260 "" 0 "" {TEXT -1 3 " " }{TEXT 266 18 "surface_int_scalar" }{TEXT -1 52 "(fcn,R,var1=rng1,var2=rng2,`step`) sis(...,`step`)" }} {PARA 260 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 268 18 "surfa ce_int_scalar" }{TEXT -1 13 "](...,`step`)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 258 17 " fcn - \+ " }{TEXT -1 51 "a scalar function of 3 variables in arrow notation " } }{PARA 0 "" 0 "" {TEXT 259 17 " R - " }{TEXT -1 89 "a para metric surface in the form of a list of 3 arrow-defined functions of t wo parameters" }}{PARA 0 "" 0 "" {TEXT 269 17 " var1,var2 - " } {TEXT -1 72 "the variables of integration and curve parameters (the or der must match)" }}{PARA 0 "" 0 "" {TEXT 270 17 " rng1,rng2 - " } {TEXT -1 46 "the ranges of the parameters to integrate over" }}{PARA 0 "" 0 "" {TEXT 260 17 " `step` - " }{TEXT -1 82 "an optional p arameter 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 -1 305 "Surface_int_scalar displays the sur face integral of a scalar field, where the first argument is the scala r field, the second argument is the surface and the third and fourth a rguments are the variables of integration and their ranges. The integ ral can then be evaluated using the value or evalf command. " }}{PARA 15 "" 0 "" {TEXT -1 108 "surface_int_scalar calculates the surface int egral of a scalar field without first displaying the integral. " }} {PARA 15 "" 0 "" {TEXT -1 259 "surface_int_scalar with the `step` para meter displays the surface integral of a scalar field and then calcula tes it while displaying all the intermediate steps. You do not need t he backquotes around `step` if the variable step has not been assigned a value. " }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part \+ of the vec_calc package, and so can be used by name only after perform ing the command with(vec_calc) or with(vec_calc,function). The functi ons can always be accessed in the long forms vec_calc[function]. The \+ aliases can be used only after performing the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "f:=makefunction([x,y,z],x^2*y+z^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&ar rowGF*,&*&9$\"\"#9%\"\"\"F3*$9&F1F3F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "R:=makefunction([theta,z],[3*cos(theta),3*sin(theta), z]); # cylinder" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%:6$%&thetaG %\"zG6\"6$%)operatorG%&arrowGF*,$-%$cosG6#9$\"\"$F*F*:F'F*F+F*,$-%$sin GF1F3F*F*:F'F*F+F*9%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Surface_int_scalar(f,R,theta=0..2*Pi,z=0..2); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,&*&-%$cosG6#%&thetaG\"\"#-%$sinG F,\"\"\"\"#\")*$%\"zGF.\"\"$/F-;\"\"!,$%#PiGF./F4;F8F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g:=MF([x,y,z],sqrt(1+x^2+y^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF**$,(\"\"\"F0*$ 9$\"\"#F0*$9%F3F0#F0F3F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S:=MF([u,v],[u*cos(v),u*sin(v),v]); # spiral ramp" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7%:6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF**&9 $\"\"\"-%$cosG6#9%F0F*F*:F'F*F+F**&F/F0-%$sinGF3F0F*F*:F'F*F+F*F4F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "surface_int_scalar(g,S, u=0..1,v=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"%\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h:=MF([x,y,z],x*z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG:6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF**&9$\"\"\"9&F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "R:=MF([theta,phi], [3*sin(phi)*cos(theta), 3*sin(phi)*sin(thet a), 3*cos(phi)]); # sphere" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7% :6$%&thetaG%$phiG6\"6$%)operatorG%&arrowGF*,$*&-%$sinG6#9%\"\"\"-%$cos G6#9$F4\"\"$F*F*:F'F*F+F*,$*&F0F4-F1F7F4F9F*F*:F'F*F+F*,$-F6F2F9F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "surface_int_scalar(h,R,th eta=0..Pi/2,phi=0..Pi/4,'step');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$-F$6$,$*(-%$cosG6#%&thetaG\"\"\"-%$sinG6#%$phiG\"\"#-F+F1F.\"# \")/F-;\"\"!,$%#PiG#F.F3/F2;F8,$F:#F.\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$*&-%'VECTORG6#7#,$*(-%$sinG6#%&thetaG\" \"\"-F06#%$phiG\"\"#-%$cosGF5F3\"#\")F3-%'MATRIXG6#7%7#&%!G6#/F2,$%#Pi G#F3F77#FA7#&FA6#/F2\"\"!F3/F6;FL,$FE#F3\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$,&*(-%$sinG6#,$%#PiG#\"\"\"\"\"#F0-F+6#% $phiGF1-%$cosGF3F0\"#\")*(-F+6#\"\"!F0F2F1F5F0!#\")/F4;F;,$F.#F0\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#,&*&-%$sinG6 #%$phiG\"\"$-F-6#,$%#PiG#\"\"\"\"\"#F6\"#F*&F,F0-F-6#\"\"!F6!#FF6-%'MA TRIXG6#7%7#&%!G6#/F/,$F4#F6\"\"%7#FD7#&FD6#/F/F " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_a liases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F:=makefunction( [x,y,z],[x*y,y+z^2,3*z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%: 6%%\"xG'%\"yG%\"zG6\"6$%)operatorG%&arrowGF+*&9$\"\"\"9%F1F+F+:F'F+F,F+ ,&F2F1*$9&\"\"#F1F+F+:F'F+F,F+,$F6\"\"$F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "R:=makefunction([theta,z],[3*cos(theta),3*sin(thet a),z]); # cylinder" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%:6$%&the taG%\"zG6\"6$%)operatorG%&arrowGF*,$-%$cosG6#9$\"\"$F*F*:F'F*F+F*,$-%$ sinGF1F3F*F*:F'F*F+F*9%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Surface_int_vector(F,R,theta=0..2*Pi,z=0..2); value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,**&-%$sinG6#%&thetaG\"\"\"-%$ cosGF,\"\"#\"#F\"\"*F.*$F/F1!\"**&F*F.%\"zGF1\"\"$/F-;\"\"!,$%#PiGF1/F 7;F;F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "G:=MF([x,y,z],[x*y,y,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%& arrowGF+*&9$\"\"\"9%F1F+F+:F'F+F,F+F2F+F+:F'F+F,F+F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S:=MF([u,v],[u*cos(v),u*sin(v),v]); # spiral ramp" }}{PARA 11 "" 1 "" {XPPM(ATH 20 "6#>%\"SG7%:6$%\"uG%\"v G6\"6$%)operatorG%&arrowGF**&9$\"\"\"-%$cosG6#9%F0F*F*:F'F*F+F**&F/F0- %$sinGF3F0F*F*:F'F*F+F*F4F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "surface_int_vector(G,S,u=0..1,v=0..Pi,`step`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,(*&%\"uG\"\"#-%$cosG6#%\"vG\"\"\"F+*& F*F+F,\"\"$!\"\"*(F*F0-%$sinGF.F0F,F0F3/F*;\"\"!F0/F/;F9%#PiG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$*&-%'VECTORG6#7#,(*&%\" uG\"\"$-%$cosG6#%\"vG\"\"\"#\"\"#F0*&F/F0F1F0#!\"\"F0*(F/F7F1F5-%$sinG F3F5#F:F7F5-%'MATRIXG6#7%7#&%!G6#/F/F57#FE7#&FE6#/F/\"\"!F5/F4;FM%#PiG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$,(-%$cosG6#%\"vG#\" \"#\"\"$*$F)F/#!\"\"F/*&F)\"\"\"-%$sinGF+F4#F2F./F,;\"\"!%#PiG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'VECTORG6#7#,(-%$sinG6#%\"vG #\"\"%\"\"**&-%$cosGF-\"\"#F+\"\"\"#!\"\"F1*$F+F5#F8F0F6-%'MATRIXG6#7% 7#&%!G6#/F.%#PiG7#FA7#&FA6#/F.\"\"!F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G,.-%$sinG6#%#PiG#\"\"%\"\"**&-%$cosGF(\"\"#F&\"\"\"#!\"\"F,*$ F&F0)#F3F+-F'6#\"\"!#!\"%F,*&-F/F7F0F6F1#F1F,*$F6F0#F1F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "H:=MF([x,y,z],[y*z,x*z,x*y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"HG7%:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+*&9%\"\"\"9&F1F+F+: F'F+F,F+*&9$F1F2F1F+F+:F'F+F,F+*&F5F1F0F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "R:=MF([theta,phi], [3*sin(phi)*cos(theta), 3*sin (phi)*sin(theta), 3*cos(phi)]); # sphere" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%:6$%&thetaG%$phiG6\"6$%)operatorG%&arrowGF*,$*&-%$sinG6# 9%\"\"\"-%$cosG6#9$F4\"\"$F*F*:F'F*F+F*,$*&F0F4-F1F7F4F9F*F*:F'F*F+F*, $-F6F2F9F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "surface_int _vector(H,R,theta=0..Pi/2,phi=0..Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$V#\"#K" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright \+ 1995-97 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 "vec_*calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "valu e" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line _int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Li ne_int_vector" 2 "Line_int_vector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " Surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ". " }}}} iquoJ+y iremJQʅ0L isqrtJPhi=0..Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$V#\"#K" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "- Copyright \+ 1995-97 by Arthur Belmonte and Philip B. Yasskin\n Department of \+ Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_*limit<linalg;"<<<<<<<<<<<<<<line?<<<====linear<lis <<=listcN<< <<<<<<<<<<<<<<<<<<====listlist<liv <<=ln <=load<local <<longg<<<<<<<<<<<<<<<<<<<<<===="\"\"*$F(\"\"'!\"\"#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *$,$*&%\"xG\"\"$,&!\"\"\"\"\"*$F&F'F*F*F)#F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&F%F%*$F$\"\"$!\"\"#F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=sqrt(x^2); simplify(f); ss(f); # incorrect if x is negative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$ *$%\"xG\"\"##\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#% \"xG\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXTlose<lower<lpmd<<<<<magnitud<main<make<<< makefunctS9<<< <<<< <<<<<<<<<==== manipulat<many<mapl<<<<<maps<match == mathematic{<<<<<<<<<<<<<<<<<<<<<<<<<<======usubsopJ%u첼 usubstringJ usystemJTP utableJ utaylorJxǰ coeffsJK convert debugoptsJʞ.h@3覃 degreeJ-R