> )*-<6=version2H&H@&8& 0&Dku+D0ִt% YO!_LxOHH-&8&8ִX&DL220&L28ִT״@t% h%&8G@&8&8ִ0&DSle2L2LE(Ķ8ƶt% YOh!1111\&8&GG8ִ0& CH{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 16 3 }{PSTYLE "Fixed \+ Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 52 " \+ Introduction to the vec_calc Package -- Version 6.0" }}{PARA 0 "" 0 " " {TEXT -1 49 "The possible HELP pages are listed at the bottom." }} {PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 17 " command(args) " }}{PARA 257 "" 0 "" {TEXT -1 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 s implify calculations which arise from vector calculus problems." }} {PARA 15 "" 0 "" {TEXT -1 49 "To load the package, be sure the system \+ variable " }{HYPERLNK 17 "libname" 2 "libname" "" }{TEXT -1 101 " incl udes the path to the directory containing the package. Then execute t he command with(vec_calc);" }}{PARA 15 "" 0 "" {TEXT -1 63 "The vec_ca lc package assumes you have also loaded 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 -1 111 "Many of the vec_calc commands have shorter aliases which become available after executing the vec_calc command " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 206 "The vec_calc commands are divided into several groups. \+ These commands are listed below by group and are followed by the alias in parentheses, if there is one. Each command is a hyperlink to its \+ help page." }}{PARA 16 "" 0 "" {TEXT -1 88 "The commands to perform fu nction definition, vector manipulation and simplification are:" }} {PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "makefunction" 2 "makef unction" "" }{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_aliases" 2 "vc_aliases" "" }{TEXT -1 37 " \+ " }}{PARA 16 "" 0 "" {TEXT -1 59 "The command s to change angular measure and coordinates 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 "CoordConversion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "p2r" 2 "CoordConversion2D" "" }{TEXT -1 5 " ) " }{HYPERLNK 17 "rect2polar" 2 "CoordConversion2D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2p" 2 "CoordConversion2D" "" }{TEXT -1 3 " ) \+ " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "cyl2rect" 2 "Coord Conversion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "c2r" 2 "CoordConver sion3D" "" }{TEXT -1 7 " ) " }{HYPERLNK 17 "rect2cyl" 2 "CoordConv ersion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "r2c" 2 "CoordConversion 3D" "" }{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 "CoordConve rsion3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "s2c" 2 "CoordConversion3 D" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "cyl2sph" 2 "CoordConversio n3D" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "c2s" 2 "CoordConversion3D" " " }{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 "curve_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 "Curv e" "" }{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 " ( " }{HYPERLNK 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 "curv e_curvature" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Ck" 2 "Cur ve" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "curve_torsion" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Ct" 2 "Curve" "" }{TEXT -1 4 " ) \+ " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "curve_tangential_ acceleration" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CaT" 2 "C urve" "" }{TEXT -1 6 " ) " }{HYPERLNK 17 "curve_normal_acceleration " 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "CaN" 2 "Curve" "" } {TEXT -1 4 " ) " }}{P ARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "cu rve_forget" 2 "Curve" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Cforget" 2 "Curve" "" }{TEXT -1 55 " ) \+ " }}{PARA 16 "" 0 "" {TEXT -1 45 "The commands for different ial 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_d eterminants" 2 "leading_principal_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_D ET" "" }{TEXT -1 8 " " }{HYPERLNK 17 "P OT" 2 "POT" "" }{TEXT -1 11 " " }{HYPERLNK 17 "VEC_POT" 2 "VEC_POT" "" }{TEXT -1 11 " " }}{PARA 16 "" 0 "" {TEXT -1 41 "The commands for inte gral 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 "mul tipleint" 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_vecto r" 2 "Line_int_vector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "Liv" 2 "Li ne_int_vector" "" }{TEXT -1 8 " ) " }{HYPERLNK 17 "line_int_vecto r" 2 "L ine_int_vector" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "liv" 2 "Li ne_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_s calar" "" }{TEXT -1 4 " ( " }{HYPERLNK 17 "sis" 2 "Surface_int_scalar " "" }{TEXT -1 3 " ) " }}{PARA 17 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "Surface_int_vector" 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 "siv" 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 cal culate the dot product of two vectors A and B, use " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc);" }}{ PARA 12 "" 1 "" {XPPMATH 20 "6#7X%#&.G%#&xG%%CURLG%$DIVG%%GRADG%)Get_VarsG%%HESSG%$JAC G%(JAC_DETG%$LAPG%0Line_int_scalarG%0Line_int_vectorG%,MuInt_noChkG%,M ultipleintG%$POTG%3Surface_int_scalarG%3Surface_int_vectorG%(VEC_POTG% &crossG%3curve_accelerationG%0curve_arclengthG%/curve_binormalG%0curve _curvatureG%-curve_forgetG%+curve_jerkG%-curve_normalG%:curve_normal_a ccelerationG%.curve_tangentG%>curve_tangential_accelerationG%.curve_to rsionG%/curve_velocityG%)cyl2rectG%(cyl2sphG%(deg2radG%$dotG%&evallG%E leading_principal_minor_determinantsG%$lenG%0line_int_scalarG%0line_in t_vectorG%-makefunctionG%,map_unapplyG%,multipleintG%+polar2rectG%(rad 2degG%)rect2cylG%+rect2polarG%)rect2sphG%(sph2cylG%)sph2rectG%#ssG%3su rface_int_scalarG%3surface_int_vectorG%+vc_aliasesG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A:=[x,y,z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "B:=[1,2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7%\"\"\"\" \"#\"\"$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(A,B);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"*&\"\"#F%%\"yGF%F%*&\"\"$ F%%\"zGF%F%" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 17 "Acknowledgements :" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 196 "The commands in th is package are used extensively throughout the text \"Multivariable Ca lculus with Maple V\" by Arthur Belmonte and Philip B. Yasskin, publis hed by Brooks/Cole under several titles:" }}{PARA 15 "" 0 "" {TEXT -1 505 "The vec_calc commands were originally written for Maple V Release 3 by A. Belmonte and P. Yasskin. The commands were organized into a \+ package for Maple V Release 3 by James Warren and P. Yasskin. The hel p pages were first written for Maple V Release 3 by David Arnold, J. W arren and P. Yasskin and converted to Maple V Release 4 by Ken Parker, Jared Teslow and P. Yasskin. The package commands were updated to Ma ple 6 and 7 by A. Belmonte and the help pages were updated to Maple 6 \+ and 7 by P. Yasskin." }}{PARA 15 "" 0 "" {TEXT -1 137 "@ Copyright 199 5-2001 by Arthur Belmonte and Philip B. Yasskin, Department of Mathema tics, Texas A&M University with all rights reserved." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "libname" 2 "li bname" "" }{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 "CoordConvers ion2D" 2 "CoordConversion2D" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "Coord Conversion3D" 2 "CoordConversion3D" "" }{TEXT -1 39 ". You are suppose d to be able to type ?" }{TEXT 258 7 "command" }{TEXT -1 14 " or ?vec_ calc[" }{TEXT 256 7 "command" }{TEXT -1 9 "] (where " }{TEXT 257 7 "co mmand" }{TEXT -1 155 " is from the above list), but this does not work for all command names. However, help is available on all commands. \+ See the Help Cross-Referencing below." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Help Cross-Referencing:" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 245 "There is a bug in the help system which prevents some \+ cross references to the help pages. So here is the list of all the he lp pages and the list of help topics which should point to each of the m. Just click on one of 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 256 "" 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 256 "" 0 "" {TEXT -1 5 " &." }}{PARA 15 "" 0 "" {HYPERLNK 17 "cross" 2 "cross" " " }}{PARA 256 "" 0 "" {TEXT -1 5 " &x" }}{PARA 15 "" 0 "" {HYPERLNK 17 "len" 2 "len" "" }}{PARA 0 "" 0 "" {TEXT -1 70 "___________________ ___________________________________________________" }}{PARA 15 "" 0 " " {HYPERLNK 17 "deg2rad" 2 "deg2rad" "" }}{PARA 256 "" 0 "" {TEXT -1 16 " d2r" }}{PARA 256 "" 0 "" {TEXT -1 16 " rad2deg r2 d" }}{PARA 15 "" 0 "" {HYPERLNK 17 "CoordConversion2D" 2 "CoordConvers ion2D" "" }}{PARA 256 "" 0 "" {TEXT -1 19 " polar2rect p2r" }} {PARA 256 "" 0 "" {TEXT -1 19 " rect2polar r2p" }}{PARA 15 "" 0 " " {HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" }}{PARA 256 "" 0 "" {TEXT -1 17 " cyl2rect c2r" }}{PARA 256 "" 0 "" {TEXT -1 17 " rect2cyl r2c" }}{PARA 256 "" 0 "" {TEXT -1 17 " sph2rect s2r" }}{PARA 256 "" 0 "" {TEXT -1 17 " rect2sph r2s" }}{PARA 256 "" 0 "" {TEXT -1 17 " sph2cyl s2c" }}{PARA 256 "" 0 "" {TEXT -1 17 " cyl2sph c2s" }}{PARA 0 "" 0 "" {TEXT -1 70 "_____________ _________________________________________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Curve" 2 "Curve" "" }}{PARA 256 "" 0 "" {TEXT -1 9 " frenet" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_velocity \+ Cv" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_accelerati on Ca" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_jerk \+ Cj" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_tangent \+ CT" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_normal \+ CN" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_binorm al CB" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_arcle ngth CL" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_curv ature Ck" }}{PARA 256 "" 0 "" {TEXT -1 36 " curve_tor sion Ct" }}{PARA 256 "" 0 "" {TEXT -1 37 " curve_ta ngential_acceleration CaT" }}{PARA 256 "" 0 "" {TEXT -1 37 " curve_ normal_acceleration CaN" }}{PARA 15 "" 0 "" {HYPERLNK 17 "curve_f orget" 2 "curve_forget" "" }}{PARA 256 "" 0 "" {TEXT -1 10 " Cforget " }}{PARA 0 "" 0 "" {TEXT -1 70 "_____________________________________ _________________________________" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Di ffops" 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 "CURL" 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 "lead ing_principal_minor_determinants" "" }}{PARA 256 "" 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 256 "" 0 "" {TEXT -1 22 " Muint" }}{PARA 256 "" 0 "" {TEXT -1 22 " multipleint muint" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Line_int_scal ar" 2 "Line_int_scalar" "" }}{PARA 256 "" 0 "" {TEXT -1 24 " \+ Lis" }}{PARA 256 "" 0 "" {TEXT -1 24 " line_int_scalar \+ lis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vec tor" "" }}{PARA 256 "" 0 "" {TEXT -1 24 " Liv" }} {PARA 256 "" 0 "" {TEXT -1 24 " line_int_vector liv" }}{PARA 15 " " 0 "" {HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }} {PARA 256 "" 0 "" {TEXT -1 27 " Sis" }}{PARA 256 "" 0 "" {TEXT -1 27 " surface_int_scalar sis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vector" "" }} {PARA 256 "" 0 "" {TEXT -1 27 " Siv" }}{PARA 256 "" 0 "" {TEXT -1 27 " surface_int_vector siv" }}{PARA 0 "" 0 " " {TEXT -1 70 "_______________________________________________________ _______________" }}}}{PAGENUMBERS 0 1 2 33 1 1 } _scalar \+ lis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vec tor" "" }}{PARA 256 "" 0 "" {TEXT -1 24 " Liv" }} {PARA 256 "" 0 "" {TEXT -1 24 " line_int_vector liv" }}{PARA 15 " " 0 "" {HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }} {PARA 256 "" 0 "" {TEXT -1 27 " Sis" }}{PARA 256 "" 0 "" {TEXT -1 27 " surface_int_scalar sis" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vector" "" }} {PARA 256 "" 0 "" {TEXT -1 27 " Siv" }}{PARA 256 "" 0 "" {TEXT -1 27 " surface_int_vector siv" }}{PARA 0 "" 0 " " {TEXT -1 70 "_______________________________________________________ _______________" }}}}{PAGENUMBERS 0 1 2 33 1 1 } Line_int_vectorSurface_int_scalarSurface_int_vector 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 vec_calc vc_aliases MF evall ssdotcrosslen {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 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"Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 P VhelpdimensionalcoordinatconversusingveccalcpackagfunctionpolarrectconvertrectangularaliascanusedafterexecutvccommandcallsequencthetaparameterhorizontalpositrightverticalupwardradialdistancoriginanglmeasurradiancounterclockwisaxisdescriptformulanorestrictvaluquadrantiviiiiiresultrangmaplarctanfunctwithargumentdesignproducexactneedthesreturnfloatpointdecimalnumberinputcontainanypartnameonlyperformalwayaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconversdegradpT xT|TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT!alia"angular4arthur belmontcallcollect coordconverscurvTdepartmdiffop5entrexpressNfunctinclud jacobian linalg:magnitud;maxneutraloutput^permit principalDreservshortsquar tangential, transformatusedvers commandveccalcvcaliassetssomepackagcallsequencdescriptdefindoneusingmaplaliaoutputincludallprevioumfmakefunctcvcurvvelocitcttangcaacceleratcnnormalcjjerkcbbinormalckcurvaturcattangentialtorscanclarclengthcforgetforgetdegradpolarrectcylsphlpmdleadprincipalminordeterminantmuintmultipleintlislineintscalarlivvectorsissurfacsivpartusedformonlyafterperformwithexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalso( functveccalcpotcalculatvectorpotentialfieldarrownotatexistcallsequencvarsparameterformlistdefinfunctionvariablnamereturnusedindependdescriptdeterminwhethgivencurltruefalsdoeswillassignsecondargummustcontainquotdiffercommandvecpotlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctevalcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopdivocity curve  elseentire   entr  %   allvalu alsog            % & ' (      alternat alwayS         % & ' (    analys analysi   angl   angular {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "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 }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 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0Ffunctionveccalcdegradconvertangldegreradianaliascanusedafterexecutvccommandcallsequencthetaparameternumbvariablexpressrepresentdescriptmeasurcontainanyfloatpointdecimalnumberreturnanswerotherwistheyexactsymbolicthespartpackagnameonlyperformwithfunctalwayaccesslongformexamplpicopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconvers`Nx )*-6 -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0  helpintroductveccalcpackagverspossiblpagelistatbottomcallsequenccommandargsdescriptcollectdesignsimplifcalculatarisvectorcalculuproblemloadsuresystemvariabllibnamincludpathdirectorcontainexecutwithassumyouhavealsostudlinalgplotmanyshortaliasbecomavailablaftervcdividintoseveralgroupthesbelowfollowaliaparentheseachhyperlinkperformfunctdefinitmanipulatsimplificatmakefunctmfevallssdotcroslenchangangularmeasurcoordinatdegradpolarrectcylsphanalyscurvvelocitcvacceleratcajerkcjtangctnormalcnbinormalcbarclengthclcurvaturcktorstangentialcatcanforgetcforgetdifferentialoperatgraddivcurllaphessleadprincipalminordeterminantlpmdjacdetpotintegralmultipleintmuintlineintscalarlislivsurfacsissivexamplproductuseacknowledgementusedextensivethroughouttextmultivariablmaplarthurbelmontphilipyasskinpublishbrookcoleundertitlwereoriginalwrittenreleasorganizjamewarrenfirstdavidarnoldconvertkenparkjaredteslowupdatcopyrightdepartmmathematictexauniversitallrightreservdiffopmultimaxmincoordconverssupposabletypeabovbutdoesworknamehowevreferencbugpreventsomeheretopicpointthemjustclickmainfrenetinalgplotmanyshortaliasbecomavailablaftervcdividintoseveralgroupthesbelowfollowaliaparentheseachhyperlinkperformfunctdefinitmanipulatsimplificatmakefunctmfevallssdotcroslenchangangularmeasurcoordinatdegradpolarrectcylsphanalyscurvvelocitcvacceleratcajerkcjtangctnormalcnbinormalcbarclengthclcurvaturcktorstangentialcatcanforgetcforgetdifferentialoperatgraddivcurllaphessleadprincipalminordeterminantlpmdjacdetpotintegralmultipleintmuintlineintscalarlislivsurfacsissivexamplproductuseacknowledgementusedextensivethroughouttextmultivariablmaplarthurbelmontphilipyasskinpublishbrookcoleundertitlwereoriginalwrittenreleasorganizjamewarrenCURL CoordConversion2D CoordConversion3D Curve DIV Diffops GRAD HESS JAC% JAC_DET& LAP Line_int_scalar* Line_int_vectorMF Multi_Max_Min Multipleint) POT' Surface_int_scalarSurface_int_vectorVEC_POT( cross curve_forget deg2raddotevall $leading_principal_minor_determinants lenss vc_aliases vec_calc  {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 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 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Cour"ier " 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 } } {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Command:" }{TEXT -1 1 " " }{TEXT 257 23 "vec_calc[vc_aliases] - " }{TEXT -1 55 "Sets aliases for some commands in the vec_calc package." }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequence: " }}{PARA 0 "" 0 "" {TEXT 256 13 " vc_aliases " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " \+ " }}{PARA 15 "" 0 "" {TEXT -1 100 "This command defines aliases for so me commands in the vec_calc package. This is done using Maple's " } {HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 56 " command. So the outp ut includes all previous aliases. " }}{PARA 15 "" 0 "" {TEXT -1 15 "Th e aliases are" }}{PARA 17 "" 0 "" {TEXT -1 10 " MF = " }{HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 49 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " Cv \+ = " }{HYPERLNK 17 "curve_velocity" 2 "Curve" "" }{TEXT -1 17 " \+ CT = " }{HY#PERLNK 17 "curve_tangent" 2 "Curve" "" }{TEXT -1 17 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " Ca = " } {HYPERLNK 17 "curve_acceleration" 2 "Curve" "" }{TEXT -1 13 " CN \+ = " }{HYPERLNK 17 "curve_normal" 2 "Curve" "" }{TEXT -1 18 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " Cj = " }{HYPERLNK 17 "curve_jerk" 2 "Curve" "" }{TEXT -1 21 " CB = " } {HYPERLNK 17 "curve_binormal" 2 "Curve" "" }{TEXT -1 16 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " Ck = " }{HYPERLNK 17 "curve_ curvature" 2 "Curve" "" }{TEXT -1 16 " CaT = " }{HYPERLNK 17 "curve_tangential_acceleration" 2 "Curve" "" }{TEXT -1 1 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " Ct = " }{HYPERLNK 17 "curve_torsion" 2 "C urve" "" }{TEXT -1 18 " CaN = " }{HYPERLNK 17 "curve_normal _acceleration" 2 "Curve" "" }{TEXT -1 5 " " }}{PARA 17 "" 0 "" {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 = " }{HYPERLNK 17 "deg2rad" 2 "deg2rad" "" }{TEXT -1 24 " r2d = " }{HYPERLNK 17 "rad2deg" 2 "deg2rad" "" } {TEXT -1 23 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " p2r = " }{HYPERLNK 17 "polar2rect" 2 "CoordConversion2D" "" } {TEXT -1 21 " r2p = " }{HYPERLNK 17 "rect2polar" 2 "Coor dConversion2D" "" }{TEXT -1 20 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " c2r = " }{HYPERLNK 17 "cyl2rect" 2 "CoordConversio n3D" "" }{TEXT -1 23 " r2c = " }{HYPERLNK 17 "rect2cyl " 2 "CoordConversion3D" "" }{TEXT -1 22 " " }} {PARA 17 "" 0 "" {TEXT -1 10 " s2r = " }{HYPERLNK 17 "sph2rect" 2 " CoordConversion3D" "" }{TEXT -1 23 " r2s = " } {HYPERLNK 17 "rect2sph" 2 "CoordConversion3D" "" }{TEXT -1 22 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " s2c = " } {HYPERLNK 17 "sph2cyl" 2 "CoordC%onversion3D" "" }{TEXT -1 24 " \+ c2s = " }{HYPERLNK 17 "cyl2sph" 2 "CoordConversion3D" "" } {TEXT -1 23 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " LPMD = " }{HYPERLNK 17 "leading_principal_minor_determinants" 2 "le ading_principal_minor_determinants" "" }{TEXT -1 25 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " Muint = " }{HYPERLNK 17 "M ultipleint" 2 "Multipleint" "" }{TEXT -1 20 " muint = " } {HYPERLNK 17 "multipleint" 2 "Multipleint" "" }{TEXT -1 19 " \+ " }}{PARA 17 "" 0 "" {TEXT -1 10 " Lis = " }{HYPERLNK 17 " Line_int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 16 " lis = \+ " }{HYPERLNK 17 "line_int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 15 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " Liv = " } {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }{TEXT -1 16 " \+ liv = " }{HYPERLNK 17 "line_int_vector" 2 "Line_int_vector" " " }{TEXT -1 15 " " }}{PARA 17 "" 0 "" {TEXT -1& 10 " Sis = " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" } {TEXT -1 13 " sis = " }{HYPERLNK 17 "surface_int_scalar" 2 "Surf ace_int_scalar" "" }{TEXT -1 12 " " }}{PARA 17 "" 0 "" {TEXT -1 10 " Siv = " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface _int_vector" "" }{TEXT -1 13 " siv = " }{HYPERLNK 17 "surface_in t_vector" 2 "Surface_int_vector" "" }{TEXT -1 12 " " }} {PARA 15 "" 0 "" {TEXT -1 138 "This command is part of the vec_calc pa ckage, and so can be used in the form vc_aliases only after performing the command with(vec_calc). " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 " Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6E%&Poin tG%#MFG%#CvG%#CaG%#CjG%#CTG%#CNG%#CBG%#CkG%#CtG%#CLG%$CaTG%$CaNG%(Cfor getG%$d2rG%$r2dG%$p2rG%$r2pG%$c2rG%$r2cG%$s2rG%$r2sG%$s2cG%$c2sG%&Muin tG%&muintG%%LPMDG%$LisG%$lisG%$LivG%$livG%$SisG%$sisG%$SivG%$sivG" }}} {EXCHG {PARA 0 "> " 0 ""' {MPLTEXT 1 0 50 "f:=MF([x,y,z],[x^2+y^3+x*y*z ^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%R6%% \"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$ F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+* &F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f:=ma kefunction([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%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arro wGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F +F,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 8 "See Also" }{TEXT -1 2 ": " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } ( ^4, y^2*z, x^2*z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%R6%% \"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$ F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+* &F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f:=ma kefunction([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%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arro wGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F +F,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 8 "See Also" }{TEXT -1 2 ": " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } (#/V b r          % & ' ( ) *  ! . 7 ; @ vec_calc vec_calc vec_calc vec_calc vec_calc vec_calc vec_calc Curve vec_calc vec_calc Diffops Diffops Diffops Diffops Diffops Diffops% Diffops& Diffops' Diffops( Diffops) vec_calc* vec_calc vec_calc vec_calc vec_calc vec_calc vec_calc vec_calc vec_calc1functionveccalclineintvectordisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandlivcallsequencvarrngparameterformlistvariablarrownotatcurvdefinparametintegratrangoveroptionalindicatdescriptfirstargumsecondthirdevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignthespartpackagnameonlyperformfunctalwayaccesslongexamplmakefunctmfcossinpicopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintscalarsurfacaliases vec_calc vectorvelocity teslow test texa{            % & ' ( ) *      text th% their them   thes?       ) *  theta:    % & they ) thinkthird *  throughout time titl top topic tors    transformat % &  you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      +.a q    E    ]  O % _& U' ( `) *   ? IJжTжЗͶ<϶ζζ4϶`ؘ TͶͶ϶ж0`Ͷxж϶lͶ$xжζжhζ|϶Ͷd϶<Ͷ϶϶жζ0жL϶Ͷжжж жHp϶϶̘HͶѶͶlж϶϶`жܗPζHжж #{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item"/ 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Command:" }{TEXT -1 1 " " }{TEXT 261 25 "vec_calc[makefunction] - " }{TEXT -1 42 "Make a Functio n using the vec_calc Package" }}{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 256 22 " MF = makefunction" }}{PARA 0 "" 0 " " {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 69 " makefunction(in,out) MF(in,out) vec_calc[makefunc tion](in,out)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 258 11 " in - " }{TEXT -1 66 "a nam e or a list of names representing the independent variable(s)" }} {PARA 0 "" 0 "" {TEXT 259 11 " out - " }{TEXT -1 112 "an expressio n or list or array of expressions representing the scalar or vector or array value of0 the function,\n" }{TEXT 260 11 " " }{TEXT -1 42 "OR nested lists and arrays of expressions." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 420 "makefunction defines a scalar- or list- or vector- or ar ray-valued function of one or more variables in arrow form. If the fu nction is list- or vector- or array-valued, then makefunction produces a list or vector or array of arrow-defined functions. If the value o f the function consists of nested lists and arrays, then makefunction \+ produces nested lists and arrays of arrow-defined functions with the s ame structure." }}{PARA 15 "" 0 "" {TEXT -1 308 "This command is part \+ of the vec_calc package, and so can be used in the form makefunction o nly after performing the command with(vec_calc) or with(vec_calc,makef unction). The command can always be accessed in the long form vec_cal c[makefunction]. The alias MF can be used only after performing the c ommand " }{HYPERLNK 17 "vc_aliases" 2 "1vc_aliases" "" }{TEXT -1 1 "." }}}{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: " }}}{PARA 0 "" 0 "" {TEXT -1 34 "A scalar function of one variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=makefunction(t,t^2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrow GF(*$)9$\"\"#\"\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 49 "A vector fu nction of one variable (e.g. a curve):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=MF(t,[t,t^2,t^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7%R6#%\"tG6\"6$%)operatorG%&arrowGF)9$F)F)F)RF'F)F*F)*$)F- \"\"#\"\"\"F)F)F)RF'F)F*F)*$)F-\"\"$F2F)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#>%\"fGR6&%\" xG%\"yG%\"zG%\"tG6\"6$%)operatorG%&arrowGF+,&*&)9$\"\"#\"\"\"-%$sinG6# *&9'F49%2F4F4F4-%#lnG6#9&F4F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 116 "A v ector function of several variables (e.g. a vector field or a coordina te transformation or a parametric surface):" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "f:=makefunction([u,v],[(u+v)/2, (u-v)/2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$R6$%\"uG%\"vG6\"6$%)operatorG% &arrowGF*,&9%#\"\"\"\"\"#*&F0F19$F1F1F*F*F*RF'F*F+F*,&F4F0*&#F1F2F1F/F 1!\"\"F*F*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 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$7%R6%%\"xG%\"yG%\"tG6\"6$%)operatorG%&arrowGF,* $)9$\"\"#\"\"\"F,F,F,RF(F,F-F,,&F2F49&F4F,F,F,RF(F,F-F,,&9%F4*&)F2\"\" $F4)F7F3F4!\"\"F,F,F,7%RF(F,F-F,F7F,F,F,RF(F,F-F,F:F,F,F,RF(F,F-F,F0F, F,F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 b y Arthur Belmonte and Philip B. Yasskin\n Department of Mathemati cs, Texas A&M University " }}{PARA 0 "" 0 "3" {TEXT 26 9 "See Also:" } {TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " Diffops" 2 "Diffops" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } "vG6\"6$%)operatorG% &arrowGF*,&9%#\"\"\"\"\"#*&F0F19$F1F1F*F*F*RF'F*F+F*,&F4F0*&#F1F2F1F/F 1!\"\"F*F*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 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$7%R6%%\"xG%\"yG%\"tG6\"6$%)operatorG%&arrowGF,* $)9$\"\"#\"\"\"F,F,F,RF(F,F-F,,&F2F49&F4F,F,F,RF(F,F-F,,&9%F4*&)F2\"\" $F4)F7F3F4!\"\"F,F,F,7%RF(F,F-F,F7F,F,F,RF(F,F-F,F:F,F,F,RF(F,F-F,F0F, F,F," }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 b y Arthur Belmonte and Philip B. Yasskin\n Department of Mathemati cs, Texas A&M University " }}{PARA 0 "" 0 "3answanswer  anti any    appear) appl  approximat arc  arclength   arctan   args argum'  ' ( ) * argument  ) aris arnold around) * arra  %  array arrowK,         % & ' ( *  arthur{            % & ' ( ) *      namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      dimens  dimensional    director displa  % display$ ) * distanc   div    (  diverg  divergenc   divid dm do) * document does' (  done   dot  dotprod doubleint) down each    )  eith elementeliminat ellips elseentire   entr  % me_'          % & ' ( ) *   named need  ) * negat   nest neutral    % & max      lineintvectorlivsurfaceintscalarsissurfaceintvectorsiv raddegdr polarrectpr cylrectsphcrs frenetcurvevelocityaccelerationjerktangentnormalbinormalarclengthcurvaturetorsiontangentialcvcacjctcncbclckcatcan cforget lpmd) multipleintmuint* lineintscalarlis makefunction {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 18 "vec_8calc[evall] - " }{TEXT -1 45 "Evaluate a List by \+ Performing Vector Algebra " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Seq uences:" }{TEXT -1 1 "\n" }{TEXT 256 40 " evall(expr) vec_calc[e vall](expr)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 257 12 " expr - " }{TEXT -1 51 "an algebraic expressio n involving lists or vectors." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 69 "evall(e xpr) 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 l9ong form vec_calc[eva ll]." }}}{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%\"\"\"\"\"#\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"vG7%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrt(2)*u + 3*v; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sqrtG6#\"\"#\"\"\"7%F)F(\"\"$F)F)7%,$%\"xGF+,$%\"yGF+,$% \"zGF+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evall(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$-%%sqrtG6#\"\"#\"\"\"F**&\"\"$F* %\"xGF*F*,&F%F)*&F,F*%\"yGF*F*,&F%F,*&F,F*%\"zGF*F*" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalm" 2 "evalm" "": }{TEXT -1 2 ", " }{HYPERLNK 17 "convert" 2 "co nvert" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "convert[listlist]" 2 "conver t[listlist]" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } }h{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"#\"\"$" }}{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#,&*&-%%sqrtG6#\"\"#\"\"\"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%,&*$-%%sqrtG6#\"\"#\"\"\"F**&\"\"$F* %\"xGF*F*,&F%F)*&F,F*%\"yGF*F*,&F%F,*&F,F*%\"zGF*F*" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalm" 2 "evalm" "": {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 15 "vec_<calc[ss] - " }{TEXT -1 41 "Symbolic Simplificatio n of an Expression " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences: " }{TEXT -1 1 "\n" }{TEXT 256 34 " ss(expr) vec_calc[ss](expr)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }{TEXT -1 1 "\n" }{TEXT 257 12 " expr - " }{TEXT -1 14 "any expression" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 " " {TEXT -1 60 "ss(expr) is entirely equivalent to simplify(expr,symbol ic). " }}{PARA 15 "" 0 "" {TEXT -1 208 "The symbolic parameter is not \+ well documented but basically it permits simplification to real answer s avoiding any complex answers. Unfortunately, it may also lose some \+ real answers. So caution is advised. " }}{PARA 15 "" 0 "" {TEXT -1 220 "This function is part of the vec_calc package, and so can be used in the form ss only after performing the command with(vec_calc) or wi th(vec_calc, ss). The function can always be accessed in the long for m vec_calc[ss]." }}}{SECT 0 {PARA 0 "" 0 "" {TE=XT 26 8 "Example:" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 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,!\"\"#F,F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),$*&)%\"xG\"\"$\"\"\",&!\"\"F**$F'F*F*F*F,#F*F)F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"),&F%F%*$)F$\"\"$F%!\"\" #F%F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=sqrt(x^2); si mplify(f); ss(f); #incorrect if x is negative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$-%%sqrtG6#*$)%\"xG\"\"#\"\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#%\"xG\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyr ight 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Departm ent of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" " " }{TEXT -1 2 ", " >}{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simplify[radical]" 2 "simplify[radical]" "" } {TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } "6#>%\"gG*$),&*$) %\"xG\"\"$\"\"\"F,*$)F*\"\"'F,!\"\"#F,F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),$*&)%\"xG\"\"$\"\"\",&!\"\"F**$F'F*F*F*F,#F*F)F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"),&F%F%*$)F$\"\"$F%!\"\" #F%F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=sqrt(x^2); si mplify(f); ss(f); #incorrect if x is negative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$-%%sqrtG6#*$)%\"xG\"\"#\"\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#%\"xG\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyr ight 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Departm ent 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 ", " > commandveccalcmakefunctmakefunctusingpackagaliacanusedafterexecutvcaliasmfcallsequencoutparameternamelistrepresentindependvariablexpressarrascalarvectorvalunestarraydescriptdefinmorearrowformproducfunctionconsistwithsamestructurpartonlyperformalwayaccesslongexamplcurvseveraldensitsinlnfieldcoordinattransformatparametricsurfaccopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffop ffunctveccalcevallevaluatlistperformvectoralgebracallsequencexprparameteralgebraicexpressinvolvdescriptentireequivalconvertevalmlistlistcalculatintomatricsimplifyresultbackpartpackagcanusedformonlyaftercommandwithalwayaccesslongexamplsqrtcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsow{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 16 "vec_K" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 258 16 "vec_calc[dot] - " }{TEXT -1 40 "Computes the Dot Prod uct of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 9 "Operator:" }{TEXT -1 1 " " }{TEXT 259 15 "vec_calc[&.] - " }{TEXT -1 40 "Computes the Do t Product of Two Vectors " }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequ ences:" }{TEXT -1 1 "\n" }{TEXT 257 45 " dot(u,v) u &. v vec _calc[dot](u,v)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 256 11 " u,v - " }{TEXT -1 36 "lists or vectors of the same length." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description :" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 73 "dot(u,v) and u &. v calculate the 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 pBerforming 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%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"*&\"\"#F%%\"bGF%F %*&\"\"$F%%\"cGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &. \+ v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"*&\"\"#F%%\"bGF%F%* &\"\"$F%%\"cGF%F%" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyrigh t 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 " See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" } {TEXT -1 2 ",C " }{HYPERLNK 17 "dotprod" 2 "dotprod" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "cross" 2 "cross" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "l en" 2 "len" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operator" 2 "operator" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "neutral" 2 "neutral" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "operators[precedence]" 2 "operators[precedence] " "" }{TEXT -1 1 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 } 2@t% 8aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dot(u ,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"*&\"\"#F%%\"bGF%F %*&\"\"$F%%\"cGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &. \+ v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"aG\"\"\"*&\"\"#F%%\"bGF%F%* &\"\"$F%%\"cGF%F%" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyrigh t 1995-2001 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 ",Cproblem   produc    product publish quadrant  quot ' ( rad    radial   radian  radical ramp rang  ) * real rect6   rectangular   recurrsive redo referenc region relat releas rememb   repeat  represent  requir#     % ' ( reserv   ' ( namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max       helpmultivariablmaxminproblemusingveccalcpackagfunctiongradcalculatgradistudequatsetequalzerouplagrangmultipliequationsolvexactcriticalpointfsolvapproximatdecimalallvaluevaluatsolutioncontainrootofhesshessianleadprincipalminordeterminantapplsecondderivattestsubsconvertsolutintolistcoordinatopstripbracketoffmakefunctmakearrowdefinfunctaliasthescanusedafterexecutvccommandlpmdmfdescriptdesignwithmultidimensionalbelowexamplbothunconstrainconstrainusenameyoumustfirstfindallclassifeachlocalmaximumminimumsaddlexpcomputdelfeqscritptdeterminnotemayfailhfmatrixatinterpretresultsincnegatpositthirdfourthfifthconfirmconcluscontourplottryrotatcontourplotorientataxesboxedabsolutvaluinsidboundarregionextremizellipsinteriorwerefoundmethodeliminatvariablconstraintnoticwenamedinsteadstillalsoupperlowerhalvhandlseparatehalfsubstitutdifferentiatdfbackrepeatfinaltabu 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {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 G"" {TEXT 26 12 "Description:" }{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%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "cross(u,v); " }}{PARA 11 "" 1 "" {XPPHMATH 20 "6#7%,&% \"cG\"\"#*&\"\"$\"\"\"%\"bGF)!\"\",&%\"aGF(F%F+,&F*F)*&F&F)F-F)F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u &x v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&%\"cG\"\"#*&\"\"$\"\"\"%\"bGF)!\"\",&%\"aGF(F%F+,&F *F)*&F&F)F-F)F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright \+ 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department o f Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "Se e Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "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 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 } 220D"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "cross(u,v); " }}{PARA 11 "" 1 "" {XPPH functveccalcsssymbolicsimplificatexpresscallsequencexprparameteranydescriptentireequivalsimplifparametwelldocumentbutbasicalpermitrealansweravoidcomplexunfortunatemayalsolosesomecautadvispartpackagcanusedformonlyafterperformcommandwithalwayaccesslongexamplsqrtincorrectnegatcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitradical`functveccalcdotcomputproductvectoroperatorcallsequencparameterlistsamelengthdescriptcalculatmodificatlinalgdotprodorthogonaloptionalwayselectpartpackagcanusednameonlyafterperformcommandwithaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocroslenneutralprecedencfunctveccalccroscalculatproductvectoroperatorcallsequenccautmustspacafterlettelsemaplwillthinksubsequletterpartnameparameterlisteachwithelementdescriptdimensionalreturnanswmodificatlinalgcrossprodinsteadpackagcanusedonlyperformcommandalwayaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodotneutralprecedencfunctveccalclencalculatlengthvectorcallsequencparameterlistdescriptmagnitudnormdefinsquarrootsumcomponentdifferlinalgpackagcomputabsolutvaluthesdiffcomplexsamealsoprevsimplificattrigidentitpartcanusedformonlyafterperformcommandwithalwayaccesslongexamplsincossimplifmaycopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitdotcalc[len] - " }{TEXT -1 35 "Calculate the Length \+ of a Vector " }}{PARA 0 "" 0 "" {TEXT 26 20 "Calling Sequences: \n" }{TEXT 256 30 " len(v) vec_calc[len](v)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 257 9 " v - " }{TEXT -1 17 "a vector or list " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT -1 153 "len(v) calculates the length or magnitude or 2-norm of t he vector v defined as the square root of the sum of the squares of th e components of the vector." }}{PARA 15 "" 0 "" {TEXT -1 364 "len(v) d iffers from norm(v,2) from the linalg package in that norm(v,2) comput es the square root of the sum of the squares of the absolute value of \+ the components of the vector. These differ if the components are comp lex so that a square is not the same as the square of the absolute val ue. The absolute values also prevent the simplification of trig ident ities." }}{PARA 15 "" 0 "" {TEXT -1 226 "The functiLon len is part of t he vec_calc package, and so can be used in the form len only after per forming the command with(vec_calc) or with(vec_calc, len). The functi on can always be accessed in the long form vec_calc[len]." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Example:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "v:=[a,b,c];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(vec_calc): len(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%% sqrtG6#,(*$)%\"aG\"\"#\"\"\"F,*$)%\"bGF+F,F,*$)%\"cGF+F,F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(linalg): norm(v,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,(*$)-%$absG6#%\"aG\"\"#\" \"\"F/*$)-F+6#%\"bGF.F/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#*$-%%sMqrtG6#,&*$)-%$sinG6#%\"tG\"\"#\"\"\"F/*$)-%$cosGF ,F.F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "norm(u,2); simplify(%); # t may be comple x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)-%$absG6#-%$sinG 6#%\"tG\"\"#\"\"\"F2*$)-F+6#-%$cosGF/F1F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)-%$absG6#-%$sinG6#%\"tG\"\"#\"\"\"F2*$ )-F+6#-%$cosGF/F1F2F2F2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Co pyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Depa rtment of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[norm]" 2 "linalg[norm]" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "dot" 2 "dot" "" }{TEXT -1 1 "." }}}} {PAGENUMBERS 0 1 2 33 1 1 } }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "len(u); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sM extensive extremiz fail fals ' ( fcn    * field+-   ' ( *  fifth final find  finish first3     % ) *  float  follow )  forget   forms?           % & ' ( ) *     formula   found fourth fr frenet   fsolv functsq        % & ' ( ) *        period permit     matric matrix    % & max       functveccalcleadprincipalminordeterminantcalculatmatrixaliacanusedafterexecutvcaliascommandlpmdcallsequencparametersquarlistexpressdescriptsubmatrictopleftcorncomputdisplayreturntheslinearalgebrapositdefinitallmultivariablcalculucriticalpointlocalminimumhessianatmaximumalternatsignbeginnwithnegatpartpackagformonlyperformalwayaccesslongexampldmsubmatrixdetcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsolinalgmultimaxminhessdeg2raddotevall frenet jerk $leading_principal_minor_determinants len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 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 Q-1 18 " r2d = rad2d eg" }}{PARA 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 Rthe vec_calc package, and so can 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 S"6#$\"+N;)R&y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rad2deg(1.); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]zdHd!\")" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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 26 2 ", " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D " "" }{TEXT 26 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConver sion3D" "" }{TEXT 26 1 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 } A 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"aG\"\"\"%#Pi G!\"\"\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "deg2rad(45); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rad2deg(Pi/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "deg2 rad(45.);" }}{PARA 11 "" 1 "" {XPPMATH 20 Scurvatur   cuts cv   cyl$   cylind  cylindrical david decimal   definS"         % & ' ( *   definit   deg    degredelf   delg   densit departm{            % & ' ( ) *         you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      ' functveccalcpotcalculatscalarpotentialvectorfieldarrownotatexistcallsequencvarsparameterformlistdefinfunctionvariablnamereturnusedindependdescriptdeterminwhethgivengraditruefalsdoeswillassignsecondargummustcontainquotdiffercommandlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctexpsingradevalcoscopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopcurl curve_forget cv cyl ddegdeg2raddotevall frenet jerk $leading_principal_minor_determinants lenlpmd makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  c{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 W0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 57 " \+ 2 Dimensional Coordinate Conversions using the vec_calc " }{TEXT -1 7 "Package" }}{PARA 0 ""X 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n " }{TEXT 256 26 " vec_calc[polar2rect] - " }{TEXT -1 46 "Converts Co ordinates from Polar to Rectangular" }}{PARA 0 "" 0 "" {TEXT 257 26 " \+ vec_calc[rect2polar] - " }{TEXT -1 46 "Converts Coordinates from Rec tangular to Polar" }}{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_aliases" "" }{TEXT -1 9 " command." }}{PARA 258 "" 0 "" {TEXT -1 21 " p2r = polar2rect" }}{PARA 259 "" 0 "" {TEXT -1 21 " r2p = rect2polar" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequ ences:" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 75 " polar2rect ([r,theta]) p2r([r,theta]) vec_calc[polar2rect]([r,theta])" }} {PARA 256 "" 0 "" {TEXT -1 71 " rect2polar([x,y]) r2p([x,y]) \+ vec_calc[rect2polar]([x,y])" }}{PARA 0 "" 0 "" {TEXT 26 12 "Param eters: " }{TEXT -1 1 "\n" }{TEXT 258 17 " [x,y] - " }{TEXT -1 25 "rectangular coordinates \n" }{TEXT 259 17 " Yx - " } {TEXT -1 50 "the horizontal coordinate, positive on the right \n" } {TEXT 260 17 " y - " }{TEXT -1 42 "the vertical coordinate , positive upward \n" }{TEXT 261 17 " [r,theta] - " }{TEXT -1 19 " polar coordinates \n" }{TEXT 262 17 " r - " }{TEXT -1 37 " the radial distance from the origin \n" }{TEXT 263 17 " theta \+ - " }{TEXT -1 72 "the angle measured in radians counterclockwise from \+ the positive x-axis " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descript ion:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 107 "polar2rect conv erts polar coordinates to rectangular coordinates using the formulas: \n " }{XPPEDIT 19 1 "x = r cos(theta);" "6#/%\"xG *&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 12 " " } {XPPEDIT 19 1 "y = r sin(theta) ;" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6#%&th etaGF'" }{TEXT -1 67 " \nThere is no restriction on the values of th e polar coordinates." }}{PARA 15 "" 0 "" {TEXT -1 138 "rect2polar conv erts rectangulaZr coordinates to polar coordinates using the formulas: \n " }{XPPEDIT 19 1 "theta = arctan(y/x);" "6#/%&thetaG-%'arctanG6#*&%\"yG\"\"\"%\"xG!\" \"" }{TEXT -1 42 " in quadrants I and IV \n " } {XPPEDIT 19 1 "r = sqrt(x^2 + y^2) ;" "6#/%\"rG-%%sqrtG6#,&*$%\"xG\"\" #\"\"\"*$%\"yGF+F," }{TEXT -1 14 " " }{XPPEDIT 19 1 "thet a = arctan(y/x) +Pi;" "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\" \"F+%#PiGF+" }{TEXT -1 71 " in quadrant II \n \+ " }{XPPEDIT 19 1 "theta=arctan(y/x )-Pi; " "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+%#PiGF-" }{TEXT -1 80 " in quadrant III \nThe resulting polar coordinates are restric ted to the ranges " }{XPPEDIT 19 1 "r >= 0" "6#1\"\"!%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 19 1 " -Pi < theta " "6#2,$%#PiG!\"\"%&thetaG" } {XPPEDIT 19 1 "``< Pi" "6#2%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 97 "Maple's arctan function with 2 ar[guments is designed to produce exactly what is needed for theta." }}{PARA 15 "" 0 "" {TEXT -1 89 "These functions return floating point decimal numbers if \+ the input contains any decimals." }}{PARA 15 "" 0 "" {TEXT -1 289 "The se functions are part of the vec_calc package, and so can be used by n ame 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 co mmand " }{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&-%$sinGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2polar([a,b]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$-%%sqrtG6#,&*$)%\"aG\"\"#\"\"\"F-* $)%\"bGF,F-F-F-\-%'arctanG6$F0F+" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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 26 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D " "" }{TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[deg2rad]" 2 "vec_calc[de g2rad]" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[rad2deg]" 2 "vec_c alc[rad2deg]" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 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&-%$sinGF)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2polar([a,b]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$-%%sqrtG6#,&*$)%\"aG\"\"#\"\"\"F-* $)%\"bGF,F-F-F-\ functveccalclapcalculatlaplacianlistfunctionarrownotatcallsequencfcnvarsparameterdefinvariablarranameusedindependdescriptargumsinglreturnformchoicoptionalrecurrsivemapsontoentrdiffercommandlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopgraddivhess cv cyl ddegdeg2raddotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity over* own packag{8            % & ' ( ) *     page  paramet#  ) *   parametero#           % & ' ( ) *    parametr  %  parametric   parametriz  parametrizat  parenthes park partk         % & ' ( ) *     partial  % path performs*          % & ' ( ) *      period permit     matric matrix    % & max      & functveccalcjacdetcalculatjacobiandeterminantcoordinattransformatcallsequencvarsparameterformvectorlistarrowdefinfunctionvariablnameusedindependdescriptmatrixreturnchoicoptionalpartpackagcanonlyafterperformcommandwithalwayaccesslongexamplmakefunctrhothetaphisincosjtsimplifcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsolinalgdiffopmuintcoordconversct curvature curve curve_forget cv cyl ddegdeg2raddotevall frenet jerk $leading_principal_minor_determinants lenlpmd makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity ) lfunctionveccalcmultipleintdisplayinertmulitiplintegralcomputmultiplpossibintermediatstepaliascanusedafterexecutvccommandmuintcallsequencxnparameterexpressintegrandeachnamerangspecifvariablintegratoptionallimitparametindicatdescriptfirstargumfollowargumentincludnumericalappearordertheyevaluatusingvalucalculatwithoutwithwhilallyoudoneedbackquotaroundassignthespartpackagonlyperformfunctalwayaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsointdoubleinttripleintjacdetlinescalarvectorsurfacfunctionveccalcsurfacintvectordisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandsivcallsequencvarrngparameterformlistvariablarrownotatparametricdefinintegratcurvrangoveroptionalparametindicatdescriptfirstargumsecondthirdfourthargumenttheirevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignappeareithorderwhatevbesthowevnamemustmatchdefinitthespartpackagonlyperformfunctalwayaccesslongexamplmakefunctthetacossincylindpimfspiralrampphisphercopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintlinescalar      _-Tlh0h^/ Ty    9{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 c }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "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 }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 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0d 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 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSeTYLE "" 0 263 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 64 " \+ 3 Dimensional Coordinate Conversions using the vec_calc Package" }} {PARA 0 "" 0 "" {TEXT 26 11 "Functions: " }{TEXT -1 1 "\n" }{TEXT 260 24 " vec_calc[cyl2rect] - " }{TEXT -1 52 "Converts Coordinates from \+ Cylindrical to Rectangular" }}{PARA 0 "" 0 "" {TEXT 261 24 " vec_cal c[rect2cyl] - " }{TEXT -1 52 "Converts Coordinates from Rectangular to Cylindrical" }}{PARAf 0 "" 0 "" {TEXT 258 24 " vec_calc[sph2rect] - \+ " }{TEXT -1 50 "Converts Coordinates from Spherical to Rectangular" }} {PARA 0 "" 0 "" {TEXT 259 24 " vec_calc[rect2sph] - " }{TEXT -1 50 " Converts Coordinates from Rectangular to Spherical" }}{PARA 0 "" 0 "" {TEXT 256 24 " vec_calc[sph2cyl] - " }{TEXT -1 50 "Converts Coordin ates from Spherical to Cylindrical" }}{PARA 0 "" 0 "" {TEXT 257 24 " \+ vec_calc[cyl2sph] - " }{TEXT -1 50 "Converts Coordinates from Cylind rical to Spherical" }}{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_aliases" "" }{TEXT -1 9 " command." }}{PARA 262 "" 0 "" {TEXT -1 19 " c2r = cyl2rect" }}{PARA 263 "" 0 "" {TEXT -1 19 " r2c = rect2cyl" }}{PARA 264 "" 0 "" {TEXT -1 19 " s2r = s ph2rect" }}{PARA 265 "" 0 "" {TEXT -1 19 " r2s = rect2sph" }} {PARA 266 "" 0 "" {TEXT -1 18 " s2c = sph2cyl" }}{PARA 267 "" 0 " " {TEXT -1 18 " c2s = cyl2sph" }}{PARA 0 "" 0 "g" {TEXT 26 18 "Call ing Sequences:" }{TEXT -1 1 " " }}{PARA 261 "" 0 "" {TEXT -1 81 " cy l2rect([r,theta,z]) c2r([r,theta,z]) vec_calc[cyl2rect]([r,the ta,z])" }}{PARA 260 "" 0 "" {TEXT -1 77 " rect2cyl([x,y,z]) \+ r2c([x,y,z]) vec_calc[rect2cyl]([x,y,z])" }}{PARA 259 "" 0 "" {TEXT -1 85 " sph2rect([rho,theta,phi]) s2r([rho,theta,phi]) vec_cal c[sph2rect]([rho,theta,phi])" }}{PARA 258 "" 0 "" {TEXT -1 77 " rect 2sph([x,y,z]) r2s([x,y,z]) vec_calc[rect2sph]([x,y,z]) " }}{PARA 257 "" 0 "" {TEXT -1 84 " sph2cyl([rho,theta,phi]) s2c([r ho,theta,phi]) vec_calc[sph2cyl]([rho,theta,phi])" }}{PARA 256 "" 0 " " {TEXT -1 80 " cyl2sph([r,theta,z]) c2s([r,theta,z]) vec_c alc[cyl2sph]([r,theta,z])" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: \+ " }}{PARA 0 "" 0 "" {TEXT 262 21 " [x,y,z] - " }{TEXT -1 24 "rectangular coordinates " }}{PARA 0 "" 0 "" {TEXT 263 21 " x \+ - " }{TEXT -1 28 "first horizontal coordinate " }}{PARA 0 "" 0 h"" {TEXT 264 21 " y - " }{TEXT -1 29 "second horizon tal coordinate " }}{PARA 0 "" 0 "" {TEXT 265 21 " z - \+ " }{TEXT -1 66 "vertical axis, positive upward and related by the righ t hand rule " }}{PARA 0 "" 0 "" {TEXT 266 21 " [r,theta,z] - " } {TEXT -1 24 "cylindrical coordinates " }}{PARA 0 "" 0 "" {TEXT 267 21 " r - " }{TEXT -1 43 "the perpendicular distance from \+ the z-axis " }}{PARA 0 "" 0 "" {TEXT 268 21 " theta - " } {TEXT -1 72 "the angle measured in radians counterclockwise from the p ositive x-axis " }}{PARA 0 "" 0 "" {TEXT 269 21 " z - \+ " }{TEXT -1 22 "same as rectangular z " }}{PARA 0 "" 0 "" {TEXT 270 21 " [rho,theta,phi] - " }{TEXT -1 22 "spherical coordinates " }} {PARA 7 "" 0 "" {TEXT -1 71 "CAUTION: The spherical coordinate system \+ used by Maple is left handed. " }}{PARA 0 "" 0 "" {TEXT 271 21 " rho - " }{TEXT -1 36 "the radial distance from the origin " } }{PARA 0 "" 0 "" {TEXT 272i 21 " theta - " }{TEXT -1 26 "sa me as cylindrical theta " }}{PARA 0 "" 0 "" {TEXT 273 21 " phi \+ - " }{TEXT -1 60 "the polar angle measured in radians from the \+ positive z-axis" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description: " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 106 "cyl2rect converts c ylindrical coordinates to rectangular coordinates using the formulas: \n " }{XPPEDIT 19 1 "x = r cos(theta);" "6#/%\"xG*&%\" rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 12 " " }{XPPEDIT 19 1 "y = r sin(theta) ;" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6#%&thetaGF'" } {TEXT -1 10 " " }{XPPEDIT 19 1 "z = z;" "6#/%\"zGF$" }{TEXT -1 73 " \nThere are no restriction on the values of the cylindrical c oordinates." }}{PARA 15 "" 0 "" {TEXT -1 142 "rect2cyl converts rectan gular coordinates to cylindrical coordinates using the formulas:\n \+ " }{XPPEDIT 19 1 "the ta = arctan(y/x);" "6#/%&thetaG-%'arctanG6#*&%\"yG\"j\"\"%\"xG!\"\"" } {TEXT -1 42 " in quadrants I and IV \n " }{XPPEDIT 19 1 "r = sqrt(x^2 + y^2) ;" "6#/%\"rG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$ %\"yGF+F," }{TEXT -1 14 " " }{XPPEDIT 19 1 "theta = arcta n(y/x) +Pi;" "6#/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+%#PiGF +" }{TEXT -1 37 " in quadrant II " }{XPPEDIT 19 1 "z = z;" "6#/%\"zGF$" }{TEXT -1 57 " \n \+ " }{XPPEDIT 19 1 "theta=arctan(y/x )-Pi;" "6 #/%&thetaG,&-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+%#PiGF-" }{TEXT -1 86 " in quadrant III \nThe resulting cylindrical coordinates are rest ricted to the ranges " }{XPPEDIT 19 1 "r >= 0;" "6#1\"\"!%\"rG" } {TEXT -1 5 " and " }{XPPEDIT 19 1 " -Pi < theta;" "6#2,$%#PiG!\"\"%&th etaG" }{XPPEDIT 19 1 "``<= Pi;" "6#1%!G%#PiG" }{TEXT -1 4 ". " }} {PARA 15 "" 0 "" {TEXT -1 97 "Maple's arctan function with 2 arguments is designed to produce exactly what is needed for theta." }}{PARA 15 "" 0 "" {TEXT -1 10k4 "sph2rect converts spherical coordinates to recta ngular coordinates using the formulas:\n " }{XPPEDIT 19 1 "x = rho * sin(phi)*cos(theta);" "6#/%\"xG*(%$rhoG\"\"\"-%$sinG6# %$phiGF'-%$cosG6#%&thetaGF'" }{TEXT -1 12 " " }{XPPEDIT 19 1 "y = rho *sin(phi) *sin(theta) ;" "6#/%\"yG*(%$rhoG\"\"\"-%$sinG6#%$ phiGF'-F)6#%&thetaGF'" }{TEXT -1 10 " " }{XPPEDIT 19 1 "z = r ho *cos(phi);" "6#/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF'" }{TEXT -1 71 " \nThere are no restriction on the values of the spherical coordinat es." }}{PARA 15 "" 0 "" {TEXT -1 127 "rect2sph converts rectangular co ordinates to spherical coordinates using the formulas:\n \+ " }{XPPEDIT 19 1 "theta = arctan(y/x);" "6#/ %&thetaG-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 28 " in quadra nts I and IV \n " }{XPPEDIT 19 1 "rho = sqrt(x^2 + y^2 + z^2) ;" "6# /%$rhoG-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yGF+F,*$%\"zGF+F," }{TEXT -1 7 " " }{XPPEDIT 19 1 "theta = arctan(ly/x) +Pi;" "6#/%&thetaG, &-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+%#PiGF+" }{TEXT -1 27 " in quad rant II " }{XPPEDIT 19 1 "phi = arccos(z/sqrt(x^2 + y^2 + z^ 2));" "6#/%$phiG-%'arccosG6#*&%\"zG\"\"\"-%%sqrtG6#,(*$%\"xG\"\"#F**$% \"yGF1F**$F)F1F*!\"\"" }{TEXT -1 44 " \n \+ " }{XPPEDIT 19 1 "theta=arctan(y/x )-Pi;" "6#/%&thetaG,&-% 'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"F+%#PiGF-" }{TEXT -1 84 " in quadran t III \nThe resulting spherical coordinates are restricted to the rang es " }{XPPEDIT 19 1 "rho >= 0;" "6#1\"\"!%$rhoG" }{TEXT -1 2 ", " } {XPPEDIT 19 1 " -Pi < theta; " "6#2,$%#PiG!\"\"%&thetaG" }{XPPEDIT 19 1 "``<= Pi;" "6#1%!G%#PiG" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "0 <= p hi; " "6#1\"\"!%$phiG" }{XPPEDIT 19 1 "``<=Pi;" "6#1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 "" 0 "" {TEXT -1 103 "sph2cyl converts spherica l coordinates to cylindrical coordinates using the formulas:\n \+ " }{XPPEDIT 19 1 "r = rho * sin(phi);" "6#/%\"rG*&%$rhoG\"\" \"m-%$sinG6#%$phiGF'" }{TEXT -1 12 " " }{XPPEDIT 19 1 "theta = theta;" "6#/%&thetaGF$" }{TEXT -1 13 " " }{XPPEDIT 19 1 "z = rho *cos(phi);" "6#/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF'" } {TEXT -1 86 " \nThere are no restriction on the values of the spheric al or cylindrical coordinates." }}{PARA 15 "" 0 "" {TEXT -1 103 "cyl2s ph converts cylindrical coordinates to spherical coordinates using the formulas:\n " }{XPPEDIT 19 1 "rho = sqrt(r^2 + z^2) ; " "6#/%$rhoG-%%sqrtG6#,&*$%\"rG\"\"#\"\"\"*$%\"zGF+F," }{TEXT -1 9 " \+ " }{XPPEDIT 19 1 "theta = theta;" "6#/%&thetaGF$" }{TEXT -1 12 " " }{XPPEDIT 19 1 "phi = arccos(z/sqrt(r^2 + z^2));" "6#/% $phiG-%'arccosG6#*&%\"zG\"\"\"-%%sqrtG6#,&*$%\"rG\"\"#F**$F)F1F*!\"\" " }{TEXT -1 140 " \nThere are no restriction on the values of the cy lindrical coordinates.\nThe resulting spherical coordinates are restri cted to the ranges " }{XPPEDIT 19 1 "rho >= 0;" "6#1\"\"!%$rhoG" } {TEXT -1 6 " and " }{XPPEDIT 19 1 "n0 <= phi; " "6#1\"\"!%$phiG" } {XPPEDIT 19 1 "``<=Pi;" "6#1%!G%#PiG" }{TEXT -1 4 ". " }}{PARA 15 " " 0 "" {TEXT -1 89 "These functions return floating point decimal numb ers if the input contains any decimals." }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part of the vec_calc package, and so can be u sed by name only after performing the command with(vec_calc) or with(v ec_calc,function). The functions can always be accessed in the long f orms vec_calc[function]. The aliases can be used only after performin g the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(vec_calc):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cyl2rect([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"aG\"\"\"-%$cosG6#%\"bGF&*&F%F&-%$sinG F)F&%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2cyl([a,b, c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$-%%sqrtG6#,&*$)%\"aG\"\"# o\"\"\"F-*$)%\"bGF,F-F-F--%'arctanG6$F0F+%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sph2rect([a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*(%\"aG\"\"\"-%$sinG6#%\"cGF&-%$cosG6#%\"bGF&*(F%F&F'F&-F(F-F& *&F%F&-F,F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rect2sph([ a,b,c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$-%%sqrtG6#,(*$)%\"aG\" \"#\"\"\"F-*$)%\"bGF,F-F-*$)%\"cGF,F-F-F--%'arctanG6$F0F+-F56$*$-F&6#, &F)F-F.F-F-F3" }}}{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%*$-%%sqrtG6 #,&*$)%\"aG\"\"#\"\"\"F-*$)%\"cGF,F-F-F-%\"bG-%'arctanG6$F+F0" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Copyright 1995-2001 by Arthur Bel monte 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" "" }p{TEXT 26 2 ", " }{HYPERLNK 17 "CoordCon version2D" 2 "CoordConversion2D" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "ve c_calc[deg2rad]" 2 "vec_calc[deg2rad]" "" }{TEXT 26 2 ", " }{HYPERLNK 17 "vec_calc[rad2deg]" 2 "vec_calc[rad2deg]" "" }{TEXT -1 2 ". " }}}} {PAGENUMBERS 0 1 2 33 1 1 } h2cyl([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%*$-%%sqrtG6 #,&*$)%\"aG\"\"#\"\"\"F-*$)%\"cGF,F-F-F-%\"bG-%'arctanG6$F+F0" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Copyright 1995-2001 by Arthur Bel monte 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" "" }p helpdimensionalcoordinatconversusingveccalcpackagfunctioncylrectconvertcylindricalrectangularsphsphericalaliascanusedafterexecutvccommandcallsequencthetarhophiparameterfirsthorizontalsecondverticalaxispositupwardrelatrighthandruleperpendiculardistancanglmeasurradiancounterclockwissamecautsystemmaplleftradialoriginpolardescriptformulanorestrictvaluquadrantiviiiiiresultrangarctanfunctwithargumentdesignproducexactneedthesreturnfloatpointdecimalnumberinputcontainanypartnameonlyperformalwayaccesslongformexamplcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocoordconversdegrad H{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 s} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 54 " \+ Frenet Analysis of a Curve using the vec_calc Package" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }}{PARA 256t "" 0 "" {TEXT -1 38 " \+ command(r) vec_calc[command](r)" }}{PARA 0 "" 0 "" {TEXT 26 11 " Parameters:" }{TEXT -1 1 "\n" }{TEXT 256 15 " command - " }{TEXT -1 43 "a command from the list below or its alias\n" }{TEXT 257 15 " \+ r - " }{TEXT -1 73 "a curve in the form of a list of arrow-de fined functions of one parameter" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 228 "These commands are \+ designed to perform a Frenet analysis of a curve. Each command has a \+ shorter alias. Most commands work in any dimension. However, the cur ve_binormal and curve_torsion commands only work in three dimensions. " }}{PARA 15 "" 0 "" {TEXT -1 95 "Below is a list of each command, the corresponding alias and a short description of its action." }}{PARA 0 "" 0 "" {TEXT 258 42 " curve_velocity Cv - " } {TEXT -1 23 "Calculate the Velocity " }}{PARA 0 "" 0 "" {TEXT 259 42 " curve_acceleration Ca - " }{TEXT -1 27 "Calculate tuh e Acceleration " }}{PARA 0 "" 0 "" {TEXT 260 42 " curve_jerk \+ Cj - " }{TEXT -1 19 "Calculate the Jerk " }}{PARA 0 " " 0 "" {TEXT 261 42 " curve_tangent CT - " } {TEXT -1 27 "Calculate the Unit Tangent " }}{PARA 0 "" 0 "" {TEXT 262 42 " curve_normal CN - " }{TEXT -1 36 "Calculat e the Unit Principal Normal " }}{PARA 0 "" 0 "" {TEXT 263 42 " curve_ binormal CB - " }{TEXT -1 38 "Calculate the Unit B inormal (3D only) " }}{PARA 0 "" 0 "" {TEXT 264 42 " curve_curvature \+ Ck - " }{TEXT -1 24 "Calculate the Curvature " }} {PARA 0 "" 0 "" {TEXT 265 42 " curve_torsion Ct \+ - " }{TEXT -1 32 "Calculate the Torsion (3D only) " }}{PARA 0 "" 0 "" {TEXT 266 42 " curve_arclength CL - " }{TEXT -1 25 "Calculate the Arc Length " }}{PARA 0 "" 0 "" {TEXT 267 42 " curve _tangential_acceleration CaT - " }{TEXT -1 38 "Calculate the Tange ntial Acceleration " }}{PARvA 0 "" 0 "" {TEXT 268 42 " curve_normal_ac celeration CaN - " }{TEXT -1 34 "Calculate the Normal Accelera tion " }}{PARA 0 "" 0 "" {TEXT 269 42 " curve_forget \+ Cforget - " }{TEXT -1 49 "Clear the remember tables for the above com mands " }}{PARA 15 "" 0 "" {TEXT -1 143 "To use the command name, you \+ must first execute with(vec_calc) or with(vec_calc,command).\nTo use t he alias, you must first execute the command " }{HYPERLNK 17 "vc_alias es" 2 "vc_aliases" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 105 "The curve must be in the form of a list of arrow-defined functions, a s can be produced using the command " }{HYPERLNK 17 "makefunction" 2 " makefunction" "" }{TEXT -1 99 " from the vec_calc package or its alias MF. Such a curve may be plotted in 2 dimensions using the " } {HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 65 " command with a parametr ic argument or in 3 dimensions using the " }{HYPERLNK 17 "spacecurve" 2 "spacecurve" "" }{TEXT -1 18 " command from the " }{HYwPERLNK 17 "plo ts" 2 "plots" "" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {TEXT -1 20 "Each command uses a " }{HYPERLNK 17 "remember" 2 "remember" "" } {TEXT -1 170 " table to speed up the computation. These tables may be \+ cleared after finishing the Frenet analysis of a curve to avoid clutte ring the memory. This is done by using the " }{HYPERLNK 17 "curve_for get" 2 "curve_forget" "" }{TEXT -1 56 " command from the vec_calc pack age or its alias Cforget." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Exa mples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): v c_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$R6#%\"tG6\"6$%) operatorG%&arrowGF)*&9$\"\"\"-%$cosG6#F.F/F)F)F)RF'F)F*F)*&F.F/-%$sinG F2F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([op(R(t) ),t=-2*Pi..2*Pi]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVxESG6$7aq7$$!3?+++3`=$G'!#<$\"3qF=n%R6[:&!#E7$$!3 mJb;*>U,?'F*$!3'>r+-ZoBD%!#=7$$!39&*=f8zn)3'F*$!3ms^fJLJ\"R)F37$$!3K8F x^GGF*7$$!3qLaUBQA+XF*$! 3MF*$!3m>$)[__z AZF*7$$!3\"ypnau&))Q7F*$!3M)yA')fNv![F*7$$!3g^ubzmklhF3$!3\"*Gjf!p!p+[ F*7$$!3gu?$!3^mF'*46i:ZF*7$$\"3yOkAkzAZdF3$!3?RCo#Q41b%F*7$ $\"3'3\"G5*))*RA6F*$!3H\"=5SseUJ%F*7$$\"3/EqBaqSN;F*$!38L?\"*4AU.SF*7$ $\"3K=XQ%H+m3#F*$!3<$Rho_d\\j$F*7$$\"3!)3E]-*H*oCF*$!3Am_'o!p.@KF*7$$ \"3'fLdx\\'zzFF*$!3IPC#45_@os>F*$\"3l$R\\ZL?V^\"F*7$$\"3OU) \\4@JOp\"F*$\"3=6b>ZKta;F*7$$\"37'R&QBl%oN\"F*$\"3?_)y26I\\w\"F*7$$\"3 !e(*eAm(eC5F*$\"3spH`\\yV:=F*7$$\"3\"e:$o%*oVRvF3$\"3q92ay8B9=F*7$$\"3 ad_ThGv*)\\F3$\"3Y^syBP$fx\"F*7$$\"3#=^\"RH(*H%G#F3$\"3]MO.1Sz*o\"F*7$ $!3Y<$)y.8O\"4)F\\t$\"3%H\"zfqsgl:F*7$$!3KbFHR.h]=F3$\"3p]&>R-#>I9F*7$ $!3]g%\\kM%H!H$F3$\"3GSi!Qs\"*oF\"F*7$$!3WJgflXWM_F3$\"3K9lZ)ehU2*F37$ $!3=!RUdab'fbF3$\"3?a))e(Q,ag&F37$$!3Kb\\O:!pR^%F3$\"3-NZ+2X\"ee#F37$$ !3!=H>*[5&zl$F3$\"3fSQ(y3-=`\"F37$$!3Wg$p!QvGGEF3$\"3-#4&H`C,dtF]p7$$! 3<=6!f/z(=?F3$\"3Q+d4znxBUF]p7$$!3![1!4=))4#Q\"F3$\"3d<`aymdT>F]p7$$!3 wdF(oSI\"psF]p$\"3mS5#R=$\\2`F\\t7$$!3J8p0@.^-iF\\t$\"3maEgrnBZQ!#A7$$ \"3a&)p'*fkg$H'F]p$\"33g8?Gl5uRF\\t7$$\"3(R?`4L'o68F3$\"3%fg#3^q*eu\"F ]p7$$\"3if%R'*>s](>F3$\"3!)3()zRQOOSF]p7$$\"3$z%=Z*)**y4EF3$\"3%e#)[Cd $zYsF]p7$$\"3`\"=1!QxL=OF3$\"3]KQH26W$\\\"F37$$\"31k(Hp8TOY%F3$\"3;rM% *emp3DF37$$\"3CqfF\\wTKbF3$\"3[3p\\Xzi\"R&F37$$\"3kh2)>%oOe_F3$\"3Cw'o (fGQ\z"**)F37$$\"35F3$\"3>- ao8g]=9F*7$$\"31')H%o!f?ZlXf3AF3$\"3') fS;R`g'o\"F*7$$!31X\\9Fa0V\\F3$\"3OV]m;'o[x\"F*7$$!3gB!>eQSck(F3$\"3^# *R^d#>]\"=F*7$$!3?\\Q.M\"=F*7$$!3mmF%*y[5r8F*$\"3k sm#Q8!\\hF*$\"38 YeFwb%*3:F*7$$!3Q;!H7E9!eAF*$\"3`()f'Q%RY;8F*7$$!3Dsa8$F*$\"3C%ocn\"HE<>F]p7$$!3;nI3L:tTKF*$!3oJ!=prc(3 TF37$$!3-J*plb*4(G$F*$!3A4b<07C$o)F37$$!3kzI>iQVnKF*$!3\\/_\")edbL8F*7 $$!3)p)Q92>o!=$F*$!3.+)37*oQ2=F*7$$!32T$>[qh`,$F*$!35A(p`81?I#F*7$$!3! o4u$RoxsFF*$!3=()z&oBhNy#F*7$$!3K#Rmp,bnY#F*$!3B#p@KN]PA$F*7$$!3IGEUk( 4?4#F*$!3U'G%)yua)HOF*7$$!3PnT$zL&yT;F*$!3#*zz&[F2*)*RF*7$$!3w,7\\`'*o H6F*$!3i1em#R*\\5VF*7$$!3+IqQ1r&>o&F3$!3gTgKGZ\"Hb%F*7$$\"3G!*>b(zMpx$ F]p$!3!o;u(z#R-s%F*7$$\"3k[0!pD3)\\iF3$!3h[VA&3C8![F*7$$\"3A$)HES{>F*$!3Ry[Z?>=>ZF*7$$\"31&))pdU5kk#F*$!3 @0dDKk^CXF*7$$\"3gUZx3#GME$F*$!3%eWe2HW%F*$!3w\"\\hsz%3ZMF*7$$\"3uB(*H]0uu\\F*$!3?&)o (\\dfb*GF*7$$\"3@#*e*=IB`?&F*$!3U\"41dKWBg#F*7$$\"3klOQNr^=aF*$!3%3I=[ 3?9H#F*7$$\"3-hZSZq*Hh&F*$!3okt^pl#Q'>F*7$$\"3Yc$Hi%3](y&F*$!3'3!RU'o+ 2i\"F*7$$\"3=X!Q*)>+9&fF*$!3+W\")Rfz0O7F*7$$\"3Idki$et%*3'F*$!3)31[w'* 4aO)F37$$\"3S:*)eS;X+iF*$!3,ZWvco-RUF37$$\"3?+++3`=$G'F*F+-%'COLOURG6& %$RGBG$\"#5!\"\"$\"\"!Ff[mFe[m-%+AXESLABELSG6$Q!6\"Fj[m-%%VIEWG6$%(DEF AULTGF_\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "curve_velocity(R );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$R6#%\"tG6\"6$%)operatorG%&arro wGF',&-%$cosG6#9$\"\"\"*&F/F0-%$sinGF.F0!\"\"F'F'F'R6#F&F'F(F',&F2F0*& F/F0F,F0F0F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "curve_a cceleration(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$R6#%\"tG6\"6$%)op eratorG%&arrowGF',&-%$sinG6#9$!\"#*&F/\"\"\"-%$cosGF.F2!\|"\"F'F'F'R6#F &F'F(F',&F3\"\"#*&F/F2F,F2F5F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "curve_jerk(R); #(The derivative of the acceleration.) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$R6#%\"tG6\"6$%)operatorG%&arrowG F',&-%$cosG6#9$!\"$*&F/\"\"\"-%$sinGF.F2F2F'F'F'R6#F&F'F(F',&F3F0*&F/F 2F,F2!\"\"F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "curve_t angent(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$R6#%\"tG6\"6$%)operato rG%&arrowGF'*&,&-%$cosG6#9$\"\"\"*&F0F1-%$sinGF/F1!\"\"F1*$-%%sqrtG6#, &F1F1*$)F0\"\"#F1F1F1F5F'F'F'RF%F'F(F'*&,&F3F1*&F0F1F-F1F1F1*$-F86#F:F 1F5F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_normal(R );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$R6#%\"tG6\"6$%)operatorG%&arro wGF',$*&,*-%$sinG6#9$\"\"#*&F.\"\"\")F1F2F4F4*(F2F4F1F4-%$cosGF0F4F4*& )F1\"\"$F4F7F4F4F4*&-%%sqrtG6#*&*$),&F2F4*$F5F4F4F2F4F4,&F4F4FDF4!\"\" F4FEF4FFFFF'F'F'RF%F'F(F',$*&,*F7!\"#*&F7F4F5F4FF*(F2F4F1F4F.F4F4*&F:F 4F.F4F4F4*&-F>6#F@F4FEF4FFFFF'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "curve_c}urvature(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#R6#%\"tG6\"6$%)operatorG%&arrowGF&*&,(*$)9$\"\"%\"\"\"F0*&F/F0)F.\" \"#F0F0F/F0F0*&),&F0F0*$F2F0F0F3F0-%%sqrtG6#*&*$),&F3F0F7F0F3F0F0F6!\" \"F0F?F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L:=curve_ar clength(R); L(0,2*Pi); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"LGR6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG6$*$-%%sqrtG6#,& \"\"\"F5*$)%\"tG\"\"#F5F5F5/F8;9$9%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#,&\"\"\"F+*$)%\"tG\"\"#F+F+F+/F.; \"\"!,$%#PiGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sqrtG6#,&\"\" \"F)*&\"\"%F))%#PiG\"\"#F)F)F)F-F)F)*&#F)F.F)-%#lnG6#,&F-!\"#*$F%F)F)F )!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "curve_tangential_ acceleration(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\"6$%)ope ratorG%&arrowGF&*&9$\"\"\"*$-%%sqrtG6#,&F,F,*$)F+\"\"#F,F,F,!\"\"F&F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "curve_normal_accelerat ion(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\"6$%)op~eratorG%&a rrowGF&*&,(*$)9$\"\"%\"\"\"F0*&F/F0)F.\"\"#F0F0F/F0F0*&,&F0F0*$F2F0F0F 0-%%sqrtG6#*&*$),&F3F0F6F0F3F0F0F5!\"\"F0F>F&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 #>%\"rG7%R6#%\"tG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"-%$cosG6#F.F/F)F) F)RF'F)F*F)*&F.F/-%$sinGF2F/F)F)F)RF'F)F*F)F.F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "spacecurve([op(r(t)),t=-2*Pi..2*Pi]);" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6#-%'CURVESG6#7T7% $!+2`=$G'!\"*$!+tjIgS!#BF(7%$!+uPiHeF*$!+xtqG:F*$!+k(Gn-'F*7%$!+9huF]F *$!+eQdJGF*$!+@AFqdF*7%$!+5f%3'RF*$!+``'e$QF*$!+yc\"Q^&F*7%$!+1fPDFF*$ !+;Ez&\\%F*$!+N\"ftD&F*7%$!+[Z*GU\"F*$!+]X?%z%F*$!+#f-4+&F*7%$!+S)p1_ \"!#5$!+H%3?u%F*$!+\\gWWZF*7%$\"+Ehr')**FT$!+SiYvVF*$!+1&*)z[%F*7%$\"+ =gCd>F*$!+#>u;v$F*$!+jH`JUF*7%$\"+8uWsEF*$!+9;mUHF*$!+?k2vRF*7%$\"+$3J l6$F*$!+JpjG?F*$!+x)>'=PF*7%$\"+3gy&G$F*$!+Ch&44\"F*$!+ML;iMF*7%$\"+A. 7*>$F*$!+XM!R0#FT$!+\"z1d?$F*7%$\"+)ej[*GF*$\"+!GYxj&FT$!+[-D\\HF*7%$ \"+)HBhU#F*$\"+(Rf$o6F*$!+0Pz#p#F*7%$\"+*3R^&=F*$\"+DHIz:F*$!+irLOCF*7 %$\"+Bg9Z7F*$\"+*fxyy\"F*$!+>1))z@F*7%$\"+]Y$Gk'FT$\"+cF20=F*$!+wSUB>F *7%$\"+3Fn+;FT$\"+kYEf;F*$!+Lv'pm\"F*7%$!+AS<^AFT$\"+b2V#R\"F*$!+!*4^5 9F*7%$!+U,UrYFT$\"+b>Gb5F*$!+ZW0a6F*7%$!+s8V'f&FT$\"+:Iq%>Q^FT$\"+4StMQFT$!+3OT6kFT7%$!+M!3dc$FT$\"+/_fV9FT$!+x\"[o%Q FT7%$!+A_vr7FT$\"+>ouR;!#6$!+YFG#G\"FT7%$\"+i^vr7FT$\"+jmuR;F_v$\"+&o# G#G\"FT7%$\"+')zqlNFT$\"+g^fV9FT$\"+;\"[o%QFT7%$\"+kT>Q^FT$\"+TRtMQFT$ \"+ZNT6kFT7%$\"+x8V'f&FT$\"+LHqG b5F*$\"+TW0a6F*7%$\"+'4u6D#FT$\"+[2V#R\"F*$\"+%)4^59F*7%$!+.En+;FT$\"+ fYEf;F*$\"+Fv'pm\"F*7%$!+@X$Gk'FT$\"+aF20=F*$\"+qSUB>F*7%$!+4g9Z7F*$\" +-w(yy\"F*$\"+81))z@F*7%$!+u!R^&=F*$\"+KHIz:F*$\"+crLOCF*7%$!+'GBhU#F* $\"+4%f$o6F*$\"+*p$z#p#F*7%$!+zN'[*GF*$\"+UkuPcFT$\"+U-D\\HF*7%$!+<.7* >$F*$!+\\K!R0#FT$\"+&y1d?$F*7%$!+3gy&G$F*$!+-h&44\"F*$\"+GL;iMF*7%$!+! 4Jl6$F*$!+4pjG?F*$\"+r)>'=PF*7%$!+EuWsEF*$!+$fhE%HF*$\"+9k2vRF*7%$!+Qg Cd>F*$!+vTn^PF*$\"+dH`JUF*7%$!+wjr')**FT$!+GiYvVF*$\"++&*)z[%F*7%$\"+a &p1_\"FT$!+C%3?u%F*$\"+VgWWZF*7%$\"+=Z*GU\"F*$!+`X?%z%F*$\"+'e-4+&F*7% $\"+wePDFF*$!+FEz&\\%F*$\"+H\"ftD&F*7%$\"+$)e%3'RF*$!+t`'e$QF*$\"+sc\" Q^&F*7%$\"+#4Yx-&F*$!+&)QdJGF*$\"+:AFqdF*7%$\"+fPiHeF*$!+6uqG:F*$\"+e( Gn-'F*7%$\"+,`=$G'F*$!+O*[F)Q!# " 0 "" {MPLTEXT 1 0 6 "Cv(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6#%\"tG6 \"6$%)operatorG%&arrowGF',&-%$cosG6#9$\"\"\"*&F/F0-%$sinGF.F0!\"\"F'F' F'R6#F&F'F(F',&F2F0*&F/F0F,F0F0F'F'F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ca(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6#%\"tG6 \"6$%)operatorG%&arrowGF',&-%$sinG6#9$!\"#*&F/\"\"\"-%$cosGF.F2!\"\"F' F'F'R6#F&F'F(F',&F3\"\"#*&F/F2F,F2F5F'F'F'\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "Cj(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6# %\"tG6\"6$%)operatorG%&arrowGF',&-%$cosG6#9$!\"$*&F/\"\"\"-%$sinGF.F2F 2F'F'F'R6#F&F'F(F',&F3F0*&F/F2F,F2!\"\"F'F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CT(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% R6#%\"tG6\"6$%)operatorG%&arrowGF',$*&,&-%$cosG6#9$!\"\"*&F1\"\"\"-%$s inGF0F4F4F4*$-%%sqrtG6#,&\"\"#F4*$)F1F " 0 "" {MPLTEXT 1 0 6 "CN(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6#%\"tG6\"6$%)operatorG%&arrowGF',$*&,*-%$sinG6#9 $\"\"%*&F.\"\"\")F1\"\"#F4F4*(\"\"$F4F1F4-%$cosGF0F4F4*&)F1F8F4F9F4F4F 4*&-%%sqrtG6#*&,(*$)F1F2F4F4*&\"\"&F4F5F4F4\"\")F4F4,&F6F4*$F5F4F4!\" \"F4FHF4FJFJF'F'F'RF%F'F(F',$*&,*F9!\"%*&F9F4F5F4FJ*(F8F4F1F4F.F4F4*&F " 0 "" {MPLTEXT 1 0 6 "CB(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6#%\"tG6\"6$%)operatorG%&arrowGF'*&,&-%$c osG6#9$!\"#*&F0\"\"\"-%$sinGF/F3F3F3*$-%%sqrtG6#,(*$)F0\"\"%F3F3*&\"\" &F3)F0\"\"#F3F3\"\")F3F3!\"\"F'F'F'RF%F'F(F',$*&,&F4FA*&F0F3F-F3F3F3*$ -F86#F:F3FCFCF'F'F'RF%F'F(F'*&,&FAF3*$F@F3F3F3*$-F86#F:F3FCF'F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ck(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\"6$%)operatorG%&arrowGF&*&,(*$)9$\"\"%\"\"\"F 0*&\"\"&F0)F.\"\"#F0F0\"\")F0F0*&),&F4F0*$F3F0F0F4F0-%%sqrtG6#*&F+F0F8 !\"\"F0F>F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ct(r);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\"6$%)operatorG%&arrowGF&*&, &*$)9$\"\"#\"\"\"F0\"\"'F0F0,(*$)F.\"\"%F0F0*&\"\"&F0F-F0F0\"\")F0!\" \"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#>%\"LGR6$%\"aG% \"bG6\"6$%)operatorG%&arrowGF)-%$IntG6$*$-%%sqrtG6#,&\"\"#\"\"\"*$)%\" tGF5F6F6F6/F9;9$9%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$* $-%%sqrtG6#,&\"\"#\"\"\"*$)%\"tGF+F,F,F,/F/;\"\"!,$%#PiGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sqrtG6#,&\"\"#\"\"\"*&\"\"%F*)%#PiGF) F*F*F*F.F*F*-%#lnG6#,&*&-F&6#F)F*F.F*F**$-F&6#,&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#R6#%\"tG6\"6$%)operatorG%&arrowGF&*&9$\"\"\"*$-%%s qrtG6#,&\"\"#F,*$)F+F2F,F,F,!\"\"F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaN(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\" 6$%)operatorG%&arrowGF&*&,(*$)9$\"\"%\"\"\"F0*&\"\"&F0)F.\"\"#F0F0\"\" )F0F0*&,&F4F0*$F3F0F0F0-%%sqrtG6#*&F+F0F7!\"\"F0F=F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_forget(r);" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "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 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 } 2F,F,F,!\"\"F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "CaN(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"tG6\" 6$%)operatorG%&arrowGF&*&,(*$)9$\"\"%\"\"\"F0*&\"\"&F0)F.\"\"#F0F0\"\" )F0F0*&,&F4F0*$F3F0F0F0-%%sqrtG6#*&F+F0F7!\"\"F0F=F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curve_forget(r);" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "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 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 }  helpfrenetanalysicurvusingveccalcpackagcallsequenccommandparameterlistbelowaliaformarrowdefinfunctionparametdescriptthesdesignperformeachshortmostworkanydimenshowevbinormaltorsonlycorrespondactionvelocitcvcalculatacceleratcajerkcjtangctunitnormalcnprincipalcbcurvaturckarclengthclarclengthtangentialcatcanforgetcforgetclearremembtablabovusenameyoumustfirstexecutwithvcaliasproducmakefunctmfsuchmayplottplotparametricargumspacecurvusesspeedupcomputatafterfinishavoidcluttermemordoneexamplcossinoppiderivatvalucopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsoaliases vec_calc velocity  ={VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 }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 27 "with(vec_calc): vc_aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=makefunction(t,[t,2*sin(t),2*cos(t)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%R6#%\"tG6\"6$%)operatorG%&arro wGF)9$F)F)F)RF'F)F*F),$-%$sinG6#F-\"\"#F)F)F)RF'F)F*F),$-%$cosGF2F3F)F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CB(r);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%R6#%\"tG6\"6$%)operatorG%&arrowGF',$*$-%%sqrtG 6#\"\"&\"\"\"#!\"#F0F'F'F'RF%F'F(F',$*&F-F1-%$cosG6#9$F1#F1F0F'F'F'RF% F'F(F',$*&F-F1-%$sinGF9F1#!\"\"F0F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cforget(r); " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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 "Curve" 2 "Curve" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makefunction" "" } {TEXT -1 1 "." }}}}{PAGENUMBERS 0 1 2 33 1 1 } 8յB 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=makefunction(t,[t,2*sin(t),2*cos(t)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%R6#%\"tG6\"6$%)operatorG%&arro wGF)9$F)F)F)RF'F)F*F), functveccalccurvforgetclearremembtablanalysialiacanusedafterexecutvcaliascommandcforgetcallsequencparametereachformlistarrowdefinfunctionparametdescriptpackagperformfrenetsuchvelocitacceleratusestortheirresultcutsdowncomputtimeotherthesallpartonlywithalwayaccesslongexamplmakefunctsincoscbcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  .{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 284 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Couri er" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 293 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 17 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 273 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 50 " \+ Differential Operstors using the vec_calc Package" }}{PARA 0 "" 0 "" {TEXT 26 9 "Functions" }{TEXT -1 2 ": " }}{PARA 266 "" 0 "" {TEXT 287 3 " " }{HYPERLNK 17 "vec_calc[makefunction]" 2 "vec_calc[makefunctio n]" "" }{TEXT 288 5 " - " }{TEXT 289 30 "Make an arrow-defined funct ion" }}{PARA 256 "" 0 "" {TEXT 261 3 " " }{HYPERLNK 17 "vec_calc[GRA D]" 2 "vec_calc[GRAD]" "" }{TEXT 262 13 " - " }{TEXT 263 22 "Calculate the Gradient" }}{PARA 259 "" 0 "" {TEXT 266 3 " " } {HYPERLNK 17 "vec_calc[DIV]" 2 "vec_calc[DIV]" "" }{TEXT 267 14 " \+ - " }{TEXT 268 24 "Calculate the Divergence" }}{PARA 260 "" 0 " " {TEXT 269 3 " " }{HYPERLNK 17 "vec_calc[CURL]" 2 "vec_calc[CURL]" "" }{TEXT 270 13 " - " }{TEXT 271 18 "Calculate the Curl" }} {PARA 261 "" 0 "" {TEXT 272 3 " " }{HYPERLNK 17 "vec_calc[LAP]" 2 "v ec_calc[LAP]" "" }{TEXT 273 14 " - " }{TEXT 274 23 "Calcula te the Laplacian" }}{PARA 257 "" 0 "" {TEXT 257 3 " " }{HYPERLNK 17 "vec_calc[HESS]" 2 "vec_calc[HESS]" "" }{TEXT 258 13 " - " } {TEXT 264 21 "Calculate the Hessian" }}{PARA 258 "" 0 "" {TEXT 259 3 " " }{HYPERLNK 17 "vec_calc[leading_principal_minor_determinants]" 2 "vec_calc[leading_principal_minor_determinants]" "" }{TEXT 260 5 " - " }{TEXT 265 50 "Calculate the Leading Principal Minor Determinants" }}{PARA 262 "" 0 "" {TEXT 275 3 " " }{HYPERLNK 17 "vec_calc[JAC]" 2 "vec_calc[JAC]" "" }{TEXT 276 14 " - " }{TEXT 277 29 "Calcu late the Jacobian Matrix" }}{PARA 263 "" 0 "" {TEXT 278 3 " " } {HYPERLNK 17 "vec_calc[JAC_DET]" 2 "vec_calc[JAC_DET]" "" }{TEXT 279 10 " - " }{TEXT 280 34 "Calculate the Jacobian Determinant" }} {PARA 264 "" 0 "" {TEXT 281 3 " " }{HYPERLNK 17 "vec_calc[POT]" 2 "v ec_calc[POT]" "" }{TEXT 282 14 " - " }{TEXT 283 36 "Find a \+ Scalar Potential if it Exists" }}{PARA 265 "" 0 "" {TEXT 284 3 " " } {HYPERLNK 17 "vec_calc[VEC_POT]" 2 "vec_calc[VEC_POT]" "" }{TEXT 285 10 " - " }{TEXT 286 36 "Find a Vector Potential if it Exists" } }{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 290 24 " \+ MF = makefunction" }}{PARA 0 "" 0 "" {TEXT 256 48 " LPMD = le ading_principal_minor_determinants" }}{PARA 0 "" 0 "" {TEXT 26 18 "Cal ling Sequences:" }{TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT -1 42 " M F(in,out) makefunction(in,out)" }}{PARA 267 "" 0 "" {TEXT -1 36 " GRAD(fcn) GRAD(fcn,vars)" }}{PARA 267 "" 0 "" {TEXT -1 33 " DIV(F) DIV(F,vars)" }}{PARA 271 "" 0 "" {TEXT -1 34 " CURL(F) CURL(F,vars)" }}{PARA 268 "" 0 "" {TEXT -1 35 " LAP(fcn) LAP(fcn,vars)" }}{PARA 272 "" 0 "" {TEXT -1 33 " LAP(F) LAP(F,vars)" }}{PARA 269 "" 0 "" {TEXT -1 36 " HESS(fcn) HESS(fcn,vars)" }}{PARA 273 "" 0 "" {TEXT -1 22 " LPMD(M) " }{TEXT 294 39 "leading_principal_ minor_determinants(M)" }}{PARA 276 "" 0 "" {TEXT -1 33 " JAC(T) \+ JAC(T,vars)" }}{PARA 275 "" 0 "" {TEXT -1 37 " JAC_DET(T) \+ JAC_DET(T,vars)" }}{PARA 270 "" 0 "" {TEXT -1 37 " POT(F,'f') \+ POT(F,'f',vars)" }}{PARA 274 "" 0 "" {TEXT -1 41 " VEC_POT(F ,'A') VEC_POT(F,'A',vars)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Paramete rs: " }}{PARA 0 "" 0 "" {TEXT 291 12 " in - " }{TEXT -1 83 "a na me or a list of names representing the independent variable(s) of the \+ function " }}{PARA 0 "" 0 "" {TEXT 292 12 " out - " }{TEXT -1 93 "an expression or list of expressions representing the scalar or vecto r value of the function " }}{PARA 0 "" 0 "" {TEXT 299 12 " fcn - \+ " }{TEXT -1 71 "a scalar field in the form of an arrow-defined functio n of n variables " }}{PARA 0 "" 0 "" {TEXT 300 12 " F - " } {TEXT -1 91 "a vector field in the form of a list or vector of n arrow -defined functions of n variables " }}{PARA 0 "" 0 "" {TEXT 301 12 " \+ " }{TEXT -1 38 "(For CURL and VEC_POT, n must be 3.) \n" } {TEXT 298 12 " M - " }{TEXT -1 50 "a square matrix or list of l ists of expressions. \n" }{TEXT 297 12 " T - " }{TEXT -1 94 "a \+ coordinate transformation in the form of a list of n arrow-defined fun ctions of k variables " }}{PARA 0 "" 0 "" {TEXT 302 12 " " }{TEXT -1 54 "(For JAC_DET, T must be square, i.e. k must equal n.) " }}{PARA 0 "" 0 "" {TEXT 295 12 " 'f' - " }{TEXT -1 49 "the name f or the scalar potential to be returned " }}{PARA 0 "" 0 "" {TEXT 293 12 " 'A' - " }{TEXT -1 49 "the name for the vector potential to b e returned " }}{PARA 0 "" 0 "" {TEXT 296 12 " vars - " }{TEXT -1 65 "a optional list of names to be used as the independent variables \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 196 "These commands are \+ designed to help with the definition, differentiation and anti-differe ntiation of scalar fields, vector fields and coordinate transformation s (including parametrized surfaces). " }}{PARA 15 "" 0 "" {TEXT -1 50 "Each command has its own help page with examples. " }}{PARA 15 "" 0 " " {TEXT -1 146 "To use the command names, you must first execute with( vec_calc) or with(vec_calc,command).\nTo use the aliases, you must fir st execute the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliases" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 19 95-2001 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 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Multi_Max_Min" 2 "Multi_Max_Min" "" }{TEXT -1 2 ". " }}} }{PAGENUMBERS 0 1 2 33 1 1 }  ect s2r -! R!ector fields and coordinate transformation s (including parametrized surfaces). " }}{PARA 15 "" 0 "" {TEXT -1 50 "Each command has its own help page with examples. " }}{PARA 15 "" 0 " " {TEXT -1 146 "To use the command names, you must first execute with( vec_calc) or with(v helpdifferentialoperstorusingveccalcpackagfunctionmakefunctmakearrowdefinfunctgradcalculatgradidivdivergenccurllaplaplacianhesshessianleadprincipalminordeterminantjacjacobianmatrixdetpotfindscalarpotentialexistvectoraliasthescanusedafterexecutvccommandmflpmdcallsequencoutfcnvarsparameternamelistrepresentindependvariablexpressvalufieldformmustsquarcoordinattransformatequalreturnoptionaldescriptdesignwithdefinitdifferentiatantiincludparametrsurfaceachownpageexampluseyoufirstcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsocurvmultimaxminsph ss tangent tangential torsion vc_aliases vec_calc velocity  if{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Times " 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 284 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 285 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 " Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 " Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 58 " \+ Multivariable Max-Min Problems using the vec_calc Package" }}{PARA 0 " " 0 "" {TEXT 26 9 "Functions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT 276 3 " " }{HYPERLNK 17 "vec_calc[GRAD]" 2 "vec_calc[GRAD]" " " }{TEXT 277 13 " - " }{TEXT 278 22 "Calculate the Gradient " }}{PARA 256 "" 0 "" {TEXT 258 3 " " }{HYPERLNK 17 "student[equate] " 2 "vec_calc[GRAD]" "" }{TEXT 259 12 " - " }{TEXT 279 74 "Se t the Gradient equal to zero or set up the Lagrange Multiplier equatio ns" }}{PARA 256 "" 0 "" {TEXT 260 3 " " }{HYPERLNK 17 "solve" 2 "sol ve" "" }{TEXT 261 22 " - " }{TEXT 280 31 "Solve for exact critical points" }}{PARA 256 "" 0 "" {TEXT 262 3 " " } {HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT 263 21 " \+ - " }{TEXT 281 45 "Solve for approximate decimal critical points" }} {PARA 256 "" 0 "" {TEXT 264 3 " " }{HYPERLNK 17 "allvalues" 2 "allva lues" "" }{TEXT 265 18 " - " }{TEXT 282 33 "Evaluate so lutions which contain " }{HYPERLNK 17 "RootOf" 2 "RootOf" "" }{TEXT 283 2 "'s" }}{PARA 256 "" 0 "" {TEXT 266 3 " " }{HYPERLNK 17 "vec_ca lc[HESS]" 2 "vec_calc[HESS]" "" }{TEXT 267 13 " - " }{TEXT 284 21 "Calculate the Hessian" }}{PARA 256 "" 0 "" {TEXT 268 3 " " } {HYPERLNK 17 "vec_calc[leading_principal_minor_determinants]" 2 "vec_c alc[leading_principal_minor_determinants]" "" }{TEXT 269 5 " - " } {TEXT 285 86 "Calculate the Leading Principal Minor Determinants to ap ply the Second Derivative Test" }}{PARA 256 "" 0 "" {TEXT 270 3 " " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 271 23 " - " }{TEXT 286 49 "Convert a solution set into a list of coordinates" } }{PARA 256 "" 0 "" {TEXT 272 3 " " }{HYPERLNK 17 "op" 2 "op" "" } {TEXT 273 25 " - " }{TEXT 287 32 "Strip brackets off a list or set" }}{PARA 256 "" 0 "" {TEXT 274 3 " " }{HYPERLNK 17 "vec_calc[makefunction]" 2 "vec_calc[makefunction]" "" }{TEXT 275 5 " - " }{TEXT 288 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 256 48 " LPMD = le ading_principal_minor_determinants" }}{PARA 0 "" 0 "" {TEXT 257 24 " \+ MF = makefunction" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Descri ption:" }}{PARA 15 "" 0 "" {TEXT -1 147 "These commands are designed t o help with multidimensional max-min problems. Below are examples of \+ both the unconstrained and constrained problems." }}{PARA 15 "" 0 "" {TEXT -1 363 "To use the command names, GRAD, HESS, leading_principal_ minor_determinants or makefunction, you must first execute with(vec_ca lc) or with(vec_calc,GRAD, HESS, leading_principal_minor_determinants, makefunction).\nTo use the command name, equate, you must first execu te with(student) or with(student, equate).\nTo use the aliases, you mu st first execute the command " }{HYPERLNK 17 "vc_aliases" 2 "vc_aliase s" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Examples : " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_ali ases:" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "The Unconstrained Probl em:" }}{PARA 0 "" 0 "" {TEXT -1 111 "Find all critical points of a fun ction and classify each as a local maximum, a local minimum or a saddl e 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#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7*&#F/\"\")F/*$)F0 F7F/F/F9F/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 solve for the critical points: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% delfG7$R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"-%$expG6#,&* $)9$\"\"#F1#!\"\"F9*&#F1\"\")F1*$)F0F9F1F1F;F1F1*(F7F1F0F1F2F1F;F*F*F* RF'F*F+F*,&*&F8F1F2F1F1*&#F1\"\"%F1*(F8F1F@F1F2F1F1F;F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqs:=equate(delf(x,y),[0,0]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,&*&%\"yG\"\"\"-%$expG6 #,&*$)%\"xG\"\"#F*#!\"\"F2*&#F*\"\")F**$)F)F2F*F*F4F*F**(F0F*F)F*F+F*F 4\"\"!/,&*&F1F*F+F*F**&#F*\"\"%F**(F1F*F9F*F+F*F*F4F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "critpts:=solve(eqs,\{x,y\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(critptsG6'<$/%\"yG\"\"!/%\"xGF)<$/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 12 "" 1 "" {XPPMATH 20 "6#-%'matri xG6#7$7$,&*(%\"xG\"\"\"%\"yGF+-%$expG6#,&*$)F*\"\"#F+#!\"\"F3*&#F+\"\" )F+*$)F,F3F+F+F5F+!\"$*()F*\"\"$F+F,F+F-F+F+,*F-F+*&F2F+F-F+F5*&#F+\" \"%F+*&F:F+F-F+F+F5**#F+FCF+F2F+F:F+F-F+F+7$F?,&F)#F;FC**#F+\"#;F+F*F+ )F,F>F+F-F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "leading_pr incipal_minor_determinants(Hf(x,y)):" }}{PARA 6 "" 1 "" {TEXT -1 37 "L eading Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"DG6#\"\"\",&*(%\"xGF'%\"yGF'-%$expG6#,&*$)F*\"\"#F'#!\"\"F2*&#F '\"\")F'*$)F+F2F'F'F4F'!\"$*()F*\"\"$F'F+F'F,F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,2*()%\"xGF'\"\"\")%\"yGF'F,)-%$expG6#,&* $F*F,#!\"\"F'*&#F,\"\")F,*$F-F,F,F6F'F,#\"\"&\"\"%*&#F,\"#;F,*(F*F,)F. F=F,F/F,F,F6*&#F,F=F,*()F+F=F,F-F,F/F,F,F6*$F/F,F6*(F'F,F*F,F/F,F,*(#F ,F'F,F-F,F/F,F,*&FFF,F/F,F6*&#F,F@F,*&FBF,F/F,F,F6" }}}{PARA 0 "" 0 " " {TEXT -1 116 "At each critical point, evaluate the Hessian and the l eading principal minor determinants and interpret the results:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p1'=p1; H1:=Hf(op(p1)); \+ LPMD(H1):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p1G7$\"\"!F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7$\"\"!\"\"\"7$F(F'" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Si nce D[2] is negative, the first critical point (0,0) is a saddle point ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p2'=p2; H2:=Hf(op(p2 )); LPMD(H2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p2G7$\"\"\"\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7$7$,$-%$expG6#!\"\"!\"%\"\" !7$F-,$F(F+" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor De terminants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$exp G6#!\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$)-%$ expG6#!\"\"F'\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Since D[2] i s 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#/%#p3G 7$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G7$7$,$-%$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 96 "S ince D[2] is positive and D[1] is positive, the third critical point ( 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$F-F(" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leadi ng Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"DG6#\"\"\",$-%$expG6#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"DG6#\"\"#,$*$)-%$expG6#!\"\"F'\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 97 "Since D[2] is positive and D[1] is positive, the fourth c ritical 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#\"\"#,$*$)-%$expG6#!\"\"F'\"\"\"\"\"%" }} }{PARA 0 "" 0 "" {TEXT -1 97 "Since D[2] is positive and D[1] is negat ive, the fifth critical point (-1,-2) is a local maximum." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "To confirm the con clusions, use a contour plot: (Try rotating the plot.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "contourplot3d(f(x,y), x=-2..2, y=-3 ..3, orientation=[-90,0], axes=boxed); " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "The Constrained Problem:" }}{PARA 0 "" 0 "" {TEXT -1 97 " Find the absolute maximum and minimum values of a function inside or o n the boundary of a region." }}{PARA 5 "" 0 "" {TEXT -1 8 "Example:" } }{PARA 0 "" 0 "" {TEXT -1 23 "Extremize the function:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=(x,y)->x*y*exp(-x^2/2-y^2/8); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7*&#F/\"\")F/*$)F0F 7F/F/F9F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 39 "inside or on the elli pse g(x,y)=1 where" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=(x, y)->x^2/4 + y^2/16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6$%\"xG% \"yG6\"6$%)operatorG%&arrowGF),&*$)9$\"\"#\"\"\"#F2\"\"%*&#F2\"#;F2)9% F1F2F2F)F)F)" }}}{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 142 "The interior critical points were found in the unco nstrained example. There are three methods of finding the critical po ints on the boundary." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 42 "Boundary Method I: Eliminate a Variable " }}{PARA 0 "" 0 "" {TEXT -1 38 "Sol ve the constraint for one variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y0:=solve(g(x,y)=1,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G6$,$*$-%%sqrtG6#,&\"\"%\"\"\"*$)%\"xG\"\"#F-!\"\"F-F1,$F'! \"#" }}}{PARA 0 "" 0 "" {TEXT -1 212 "Notice that we named the solutio n y0 instead of y so that we can still use y as a variable. Also noti ce that there are 2 solutions for the upper and lower halves of the el lipse. We must handle these separately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the upper half of the boundary , substitute the solution into the function and differentiate:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f1:=makefunction(x,f(x,y0[1] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6$%)operatorG% &arrowGF(,$*(9$\"\"\"-%%sqrtG6#,&\"\"%F/*$)F.\"\"#F/!\"\"F/-%$expG6#! \"#F/F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Df1:=D(f1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df1GR6#%\"xG6\"6$%)operatorG%& arrowGF(,&*&-%%sqrtG6#,&\"\"%\"\"\"*$)9$\"\"#F3!\"\"F3-%$expG6#!\"#F3F 7*&*(F7F3F5F3F9F3F3*$-F/6#F1F3F8F8F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 90 "Find the x-coordinate at each critical point and substitute back t o find the y-coordinate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " x0:=solve(Df1(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$-% %sqrtG6#\"\"#\"\"\",$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,$*$-%%sqrtG6#\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$-%%sqrtG6#\"\"#\"\"\",$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y2:=subs(x=x0[2],y0[1]); b2:=[x0[2],y2]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*$-%%sqrtG6#\"\"#\"\"\"F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$,$*$-%%sqrtG6#\"\"#\"\"\"! \"\",$F'F+" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Repeat for the lower half \+ of the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f2:=make function(x,f(x,y0[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR6#% \"xG6\"6$%)operatorG%&arrowGF(,$*(9$\"\"\"-%%sqrtG6#,&\"\"%F/*$)F.\"\" #F/!\"\"F/-%$expG6#!\"#F/F " 0 "" {MPLTEXT 1 0 11 "Df2:=D(f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df2 GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&-%%sqrtG6#,&\"\"%\"\"\"*$)9$\" \"#F3!\"\"F3-%$expG6#!\"#F3F<*&*(F7F3F5F3F9F3F3*$-F/6#F1F3F8F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df2(x)=0,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$-%%sqrtG6#\"\"#\"\"\",$F&! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y3:=subs(x=x0[1],y0 [2]); b3:=[x0[1],y3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G,$*$-%% sqrtG6#\"\"#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$*$-% %sqrtG6#\"\"#\"\"\",$F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y4:=subs(x=x0[2],y0[2]); b4:=[x0[2],y4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y4G,$*$-%%sqrtG6#\"\"#\"\"\"!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#b4G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'!\"#" }}} {PARA 0 "" 0 "" {TEXT -1 139 "Finally, we tabulate the values of the f unction at all interior and boundary 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$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$e xpG6#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+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(p 4)); 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 "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#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1; f(op(b1)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*$-%%sqrtG6#\"\"#\"\"\",$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(b 2)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$-%%sqrtG6#\"\" #\"\"\"!\"\",$F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#! \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b3; f(op(b3)); evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$*$-%%sqrtG6#\"\"#\"\"\",$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(b 4)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$-%%sqrtG6#\"\" #\"\"\"!\"\",$F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\" #\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8T8a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 154 "So we see that the absolute maxima occur at the int erior points (1,2) and (-1,-2), and the absolute minima occur at the i nterior points (1,-2) and (-1,2)." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 47 "Boundary Method II: Parametrize the Boundary " }}{PARA 0 "" 0 "" {TEXT -1 27 "Define the parametrization:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r:=makefunction(t,[2*cos(t),4*sin(t)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7$R6#%\"tG6\"6$%)operatorG%&arro wGF),$-%$cosG6#9$\"\"#F)F)F)RF'F)F*F),$-%$sinGF0\"\"%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Restrict the function to the boundary:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fr:=makefunction(t,simplify( f(op(r(t)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#frGR6#%\"tG6\"6$% )operatorG%&arrowGF(,$*(-%$cosG6#9$\"\"\"-%$sinGF0F2-%$expG6#!\"#F2\" \")F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Find the critical points on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Dfr:=D(fr) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DfrGR6#%\"tG6\"6$%)operatorG%& arrowGF(,&*&)-%$sinG6#9$\"\"#\"\"\"-%$expG6#!\"#F4!\")*(\"\")F4)-%$cos GF1F3F4F5F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bndc ritpts:=solve(Dfr(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+bndcr itptsG6$,$%#PiG#\"\"\"\"\"%,$F'#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b1:=r(bndcritpts[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$-%%sqrtG6#\"\"#\"\"\",$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b2:=r(bndcritpts[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$-%%sqrtG6#\"\"#\"\"\",$F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Since the equation is non-polynomial, solve may not give all solutions. So we plot the function for one period:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(Dfr,-Pi..Pi);" }}} {PARA 0 "" 0 "" {TEXT -1 85 "From the plot and its symmetries, it is o bvious that solve missed two more solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b3:=r(3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#b3G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b4:=r(-3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 206 "Finally, we tabulate the values of the function at all i nterior and boundary critical points and identify the absolute maximum and minimum: \n(This was done with the first method. So we won't red o it here.)" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 49 "Boundary Method I II: Lagrange Multiplier Method" }}{PARA 0 "" 0 "" {TEXT -1 69 "Find \+ the gradient of the function and the gradient of the constraint:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%delfG7$R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF *,&*&9%\"\"\"-%$expG6#,&*$)9$\"\"#F1#!\"\"F9*&#F1\"\")F1*$)F0F9F1F1F;F 1F1*(F7F1F0F1F2F1F;F*F*F*RF'F*F+F*,&*&F8F1F2F1F1*&#F1\"\"%F1*(F8F1F@F1 F2F1F1F;F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delg:=GRA D(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG7$R6$%\"xG%\"yG6\"6$% )operatorG%&arrowGF*,$9$#\"\"\"\"\"#F*F*F*RF'F*F+F*,$9%#F1\"\")F*F*F* " }}}{PARA 0 "" 0 "" {TEXT -1 91 "Set up the Lagrange multiplier equat ions and solve for the critical points on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqs:=equate(delf(x,y),lambda*delg(x ,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,&*&%\"yG\"\"\"-%$e xpG6#,&*$)%\"xG\"\"#F*#!\"\"F2*&#F*\"\")F**$)F)F2F*F*F4F*F**(F0F*F)F*F +F*F4,$*&%'lambdaGF*F1F*#F*F2/,&*&F1F*F+F*F**&#F*\"\"%F**(F1F*F9F*F+F* F*F4,$*&F=F*F)F*#F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "so l:=solve(\{op(eqs),g(x,y)=1\},\{x,y,lambda\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG6$<%/%\"yG,$-%'RootOfG6$,&*$)%#_ZG\"\"#\"\"\"F2F 1!\"\"/%&labelG%$_L5GF1/%\"xGF*/%'lambdaG,$-%$expG6#!\"#!\"%<%F'/F8,$F *F3/F:,$F<\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Since the solutions i nvolve a RootOf, we resolve them using allvalues:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=allvalues(sol[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G6$<%/%'lambdaG,$-%$expG6#!\"#!\"%/%\"xG*$-%%sqr tG6#\"\"#\"\"\"/%\"yG,$F1F5<%F'/F8,$F1F-/F0,$F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1:=subs(sol1[1],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$-%%sqrtG6#\"\"#\"\"\",$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b4:=subs(sol1[2],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'! \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol2:=allvalues(sol[ 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G6$<%/%'lambdaG,$-%$exp G6#!\"#\"\"%/%\"xG,$*$-%%sqrtG6#\"\"#\"\"\"!\"\"/%\"yG,$F2F6<%F'/F:,$F 2F-/F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b3:=subs(sol2[1 ],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$,$*$-%%sqrtG6#\" \"#\"\"\"!\"\",$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2: =subs(sol2[2],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$-%% sqrtG6#\"\"#\"\"\",$F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 208 "Finally, \+ we tabulate the values of the function at all interior and boundary cr itical points and identify 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 -1 117 "- Copyright 1995-2001 by Arthur B elmonte 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 "GRAD" 2 "GRAD" "" }{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 "leadin g_principal_minor_determinants" 2 "leading_principal_minor_determinant s" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "op" 2 "op" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefu nction" 2 "makefunction" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } XT -1 2 ", " } {HYPERLNK 17 "GRAD" 2 "GRAD" "" }{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 ", " } {HYPmaxima maximum  may   measur   memor method mf+   *    min      minima minimum  minor&      miss  modificat more   most muint'% & ) *   mulitipl) multi     multidimensional multipl)  multipleint)  multipli  multivariabl   must#   ' ( namec*          % & ' ( ) *   named need  ) * negat   nest neutral % & max      no   non normnormal   notat/      ' ( * note notic numbnumber   numerical) obviou occur off onlyo(          % & ' ( ) *     onto op   operat operator operstor optionoptional7      % & ) * order) organiz orientat origin   original  orthogonalother otherwisout   output       matric matrix    % & max      resolv respect% restrict   result     return?         % & ' ( rho % & right   rng* rootrootof rotat rule saddl same  scalar3T   ' ) *    second'   ' ( * selectseparate sequencw           % & ' ( ) *      set sets several    short    ' ( ) *   named need  ) * negat   nest neutral % & max      function[5           % & ' ( ) *  give given ' ( grad'"       '  gradi   ' group half halv hand handl have help      here   hess      hessian    hf  hg  horizontal   howev   hyperlink identif identitii   iii   ij% includ )    orthogonalother otherwisout   output * negat   nest neutral    % & max       perpendicular phi % & philip{            % & ' ( ) *      pi  * plot   plott point     polar     polynomial posit    possib) * possibl pot/    ' (   potential ' (  precedenc prevprevent previou pricipal  principal&      namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      latidentifevalfmaximaoccurminimaiiparametrizparametrizatcossinrestrictfrsimplifdfrbndcritptnonpolynomialgiveperiodpisymmetrobvioumissmoredonewonredohereiiidelglambdasolinvolvresolvthemcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitcondderivattestsubsconvertsolutintolistcoordinatopstripbracketoffmakefunctmakearrowdefinfunctaliasthescanusedafterexecutvccommandlpmdmfdescriptdesignwithmultidimensionalbelowexamplbothunconstrainconstrainusenameyoumustfirstfindallclassifeachlocalmaximumminimumsaddlexpcomputdelfeqscritptdeterminnotemayfailhfmatrixatinterpretresultsincnegatpositthirdfourthfifthconfirmconcluscontourplottryrotatcontourplotorientataxesboxedabsolutvaluinsidboundarregionextremizellipsinteriorwerefoundmethodeliminatvariablconstraintnoticwenamedinsteadstillalsoupperlowerhalvhandlseparatehalfsubstitutdifferentiatdfbackrepeatfinaltabu {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item " 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {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#>%\"fGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF**()9$ \"\"#\"\"\")9%\"\"$F2)9&\"\"%F2F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% delfG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,$*(9$\"\"\")9%\"\" $F2)9&\"\"%F2\"\"#F+F+F+RF'F+F,F+,$*()F1F9F2)F4F9F2F6F2F5F+F+F+RF'F+F, F+,$*(F=F2F3F2)F7F5F2F8F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "delf(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(%\"xG\"\" \")%\"yG\"\"$F')%\"zG\"\"%F'\"\"#,$*()F&F.F')F)F.F'F+F'F*,$*(F1F'F(F') F,F*F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=makefunction ([x,y],2*x^2*y+exp(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6$% \"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&)9$\"\"#\"\"\"9%F2F1-%$expG6#F3 F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "delg:=GRAD(g,[a ,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delgG7$R6$%\"aG%\"bG6\"6$% )operatorG%&arrowGF*,$*&9$\"\"\"9%F1\"\"%F*F*F*RF'F*F+F*,&*$)F0\"\"#F1 F8-%$expG6#F2F1F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "de lg(p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&%\"pG\"\"\"%\"qGF'\" \"%,&*$)F&\"\"#F'F--%$expG6#F(F'" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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[g rad]" "" }{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 "DIV" 2 "DIV" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } eratorG%&arrowGF*,$*&9$\"\"\"9%F1\"\"%F*F*F*RF'F*F+F*,&*$)F0\"\"#F1 F8-%$expG6#F2F1F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "de lg(p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&%\"pG\"\"\"%\"qGF'\" \"%,&*$)F&\"\"#F' functveccalcgradcalculatgradiarrownotatcallsequencfcnvarsparameterscalarseveralvariabllistnameusedindependdescriptvectorfirstpartialderivatactsdefinreturnfunctionchoicoptionaldiffercommandlinalgpackagexpressrequirspecificatpartcanformonlyafterperformwithalwayaccesslongexampldelfmakefunctexpdelgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopmultimaxmindivcurlhesspotdeg2raddotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  e{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 }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%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG F+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F ,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+*&F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "DIV(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF(,*9$\"\"#*&9%\"\"\")9&\"\"% F1F1*(F.F1F0F1F3F1F1*(\"\"$F1)F-F.F1)F3F.F1F1F(F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and P hilip B. Yasskin\n Department of Mathematics, Texas A&M Universit y " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[diverge]" 2 "linalg[diverge]" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CURL" 2 "CURL" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "VEC_POT" 2 "VEC_POT" "" }{TEXT -1 2 ". \+ " }}}}{PAGENUMBERS 0 1 2 33 1 1 }  @UseHardwareFloats"6$%)operatorG%&arrowG F+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F ,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+*&F1 functveccalcdivcalculatdivergencvectorfieldarrownotatcallsequencvarsparameterdimensionalformlistdefinfunctionvariablnameusedindependdescriptreturnchoicoptionaldiffercommanddiverglinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopgradcurlpot cyl ddegdeg2raddotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item " 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {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%R6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\"\"$F4F4*(F2F4F7F4) 9&\"\"%F4F4F+F+F+RF'F+F,F+*&)F7F3F4F;F4F+F+F+RF'F+F,F+*&F1F4)F;F8F4F+F +F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CURL(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF), $*$)9%\"\"#\"\"\"!\"\"F)F)F)RF%F)F*F),&*(9$F2F0F2)9&\"\"$F2\"\"%*(F1F2 F7F2F8F2F3F)F)F)RF%F)F*F),&F.!\"$*&F7F2)F9F;F2F3F)F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M Uni versity " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " } {HYPERLNK 17 "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 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } &菺R'F+F,F+*&) functveccalccurlcalculatdimensionalvectorfieldarrownotatcallsequencvarsparameterformlistdefinfunctionvariablnameusedindependdescriptreturnchoicoptionaldiffercommandlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformwithalwayaccesslongexamplmakefunctcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopgraddivpot curve_forget cv cyl ddegdeg2raddotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Outpu t" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } } {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%R 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(*$)9$\"\"#\"\"\"F4*$)9%\" \"$F4F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F,F+*&)F7F3F4F;F4F+F+F+RF'F+F, F+*&F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "LAP (F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6%%\"xG%\"yG%\"zG6\"6$%)ope ratorG%&arrowGF),(\"\"#\"\"\"*&\"\"'F/9%F/F/**\"#7F/9$F/F2F/)9&F.F/F/F )F)F)RF%F)F*F),$F7F.F)F)F)RF%F)F*F),&*$)F7\"\"$F/F.*(F1F/)F5F.F/F7F/F/ F)F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathema tics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "linalg[laplacian]" 2 "linalg[laplacian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DIV" 2 "DIV" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "HESS" 2 "HESS" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } ,h !F4*(F2F4F7F4)9&\"\"%F4F4F+F+F+RF'F+F,F+*&)F7F3F4F;F4F+F+F+RF'F+F, F+*&F1F4)F;F8F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "LAP (F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6%%\"xG%\"yG%\"zG6\"6$%)ope ratorG%&arrowGF),(\"\"#\"\"\"*&\"\"'F/9%F/F/**\"#7F/9$F/F2F/)9&F.F/F/F )F)F)RF%F)F*F),$F7F.F)F)F)RF%F)F*F),&*$)F7\"\"$F/F.*(F1F/)F5F.F/F7F/F/ F)F)F)" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathema tics,below   best binormal    bndcritpt both bottom boundar boxed bracket brook bug but   ca   calc{0       % & ' ( ) *     calculat_3        % & ' ( ) *   calculu   callw           % & ' ( ) *       origin   original  orthogonalother otherwisout   output * negat   nest neutral    % & max      can{Z            % & ' ( ) *      cat   caut  cb    cforget    chang choic     % & cj   ck   cl   classif clear   click clutter cn   cole collect     tangential    % & previou pricipal  principal&              % & ' ( ) *    eqs equal   equat equation equival   eval ' ( evalf  * evall  evalm evaluat  ) *  exact   exampl{$            % & ' ( ) *      execut?       ) *   exist ' ( exp   ' expr  express?       % ' ( )    fsolv functsq        % & ' ( ) *        period permit     matric matrix    % & max       g{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 17 "vec_calc[HESS] - " }{TEXT -1 55 "Calculates the Hessi an of a Function in Arrow Notation " }}{PARA 0 "" 0 "" {TEXT 26 20 "Ca lling Sequences: \n" }{TEXT 256 41 " HESS(fcn) vec_calc[HES S](fcn)" }}{PARA 0 "" 0 "" {TEXT 258 46 " HESS(fcn,vars) vec_cal c[HESS](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 names 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 Hessian of a function is the matrix of second parti al derivatives of the function." }}{PARA 15 "" 0 "" {TEXT -1 212 "HESS acts on an arrow-defined function and returns a list of lists of the \+ second partial derivatives as arrow defined functions. The choice of \+ variables is optional. To display the Hessian as an array, use the " }{HYPERLNK 17 "matrix" 2 "matrix" "" }{TEXT -1 9 " command." }}{PARA 15 "" 0 "" {TEXT -1 30 "HESS differs from the command " }{HYPERLNK 17 "hessian" 2 "hessian" "" }{TEXT -1 8 " in the " }{HYPERLNK 17 "linalg " 2 "linalg" "" }{TEXT -1 133 " package:\nlinalg[hessian] acts on an e xpression and returns a matrix of expressions. It requires the specif ication of the variables." }}{PARA 15 "" 0 "" {TEXT -1 230 "The functi on HESS is part of the vec_calc package, and so can be used in the for m HESS only after performing the command with(vec_calc) or with(vec_ca lc, HESS). The function can always be accessed in the long form vec_c alc[HESS]." }}}{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#>%\"fGR6%%\"xG%\"yG%\"zG6\"6$%)ope ratorG%&arrowGF**()9$\"\"#\"\"\")9%\"\"$F2)9&\"\"%F2F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Hf:=HESS(f);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#HfG7%7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF,, $*&)9%\"\"$\"\"\")9&\"\"%F5\"\"#F,F,F,R6%F)F*F+F,F-F,,$*(9$F5)F3F9F5F6 F5\"\"'F,F,F,R6%F)F*F+F,F-F,,$*(F>F5F2F5)F7F4F5\"\")F,F,F,7%R6%F)F*F+F ,F-F,FF9F5F3F5F6F5F@F,F,F,R6%F)F*F+F,F-F, ,$*(FNF5F?F5FEF5\"#7F,F,F,7%R6%F)F*F+F,F-F,FCF,F,F,R6%F)F*F+F,F-F,FQF, F,F,R6%F)F*F+F,F-F,,$*(FNF5F2F5)F7F9F5FSF,F,F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "Hf(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%7%,$*&)%\"yG\"\"$\"\"\")%\"zG\"\"%F*\"\"#,$*(%\"xGF*)F(F.F*F+F*\"\"' ,$*(F1F*F'F*)F,F)F*\"\")7%F/,$*()F1F.F*F(F*F+F*F3,$*(F;F*F2F*F6F*\"#77 %F4F<,$*(F;F*F'F*)F,F.F*F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "matrix(Hf(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6# 7%7%,$*&)%\"yG\"\"$\"\"\")%\"zG\"\"%F-\"\"#,$*(%\"xGF-)F+F1F-F.F-\"\"' ,$*(F4F-F*F-)F/F,F-\"\")7%F2,$*()F4F1F-F+F-F.F-F6,$*(F>F-F5F-F9F-\"#77 %F7F?,$*(F>F-F*F-)F/F1F-FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=makefunction([x,y],2*x^2*y+exp(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&)9$\" \"#\"\"\"9%F2F1-%$expG6#F3F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Hg:=HESS(g,[a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#HgG7$7$R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF+,$9%\"\"%F+F+F+R6$F) F*F+F,F+,$9$F1F+F+F+7$R6$F)F*F+F,F+F4F+F+F+R6$F)F*F+F,F+-%$expG6#F0F+F +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 -1 117 "- Copyright 1 995-2001 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[hessian]" 2 "linalg[hessian]" " " }{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 "GRAD" 2 "GRAD" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "LA P" 2 "LAP" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "vec_calc[leading_pricipa l_minor_determinants]" 2 "vec_calc[leading_pricipal_minor_determinants ]" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } 3PD! &(+F4F+F+F+R6$F)F*F+F,F+-%$expG6#F0F+F +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 -1 117 "- Copyright 1 995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" functveccalchesscalculathessianarrownotatcallsequencfcnvarsparameterscalarseveralvariabllistnameusedindependdescriptmatrixsecondpartialderivatactsdefinreturnfunctionchoicoptionaldisplaarrausecommanddifferlinalgpackagexpressrequirspecificatpartcanformonlyafterperformwithalwayaccesslongexamplhfmakefunctexphgcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopmultimaxmingradlapleadpricipalminordeterminantddegdeg2raddotevall frenet jerk len makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity  x{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 259 49 "vec_calc[leading_principal_minor_determinants] - " } {TEXT -1 63 "Calculates the Leading Principal Minor Determinants of a \+ Matrix" }}{PARA 0 "" 0 "" {TEXT 26 6 "Alias:" }{TEXT -1 48 " - The ali as can be used after execution of the " }{HYPERLNK 17 "vc_aliases" 2 " vc_aliases" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT 257 48 " LPMD = leading_principal_minor_determinants" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 258 115 " leading_principal_minor_determinants(M) \+ LPMD(M)\n vec_calc[leading_principal_minor_determinants](M)" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT 256 9 " M - " }{TEXT -1 49 "a square matrix or list o f lists of expressions. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Desc ription:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 35 "The leading \+ principal minors of an " }{TEXT 260 5 "n x n" }{TEXT -1 23 " square ma trix are the " }{TEXT 261 5 "k x k" }{TEXT -1 47 " square submatrices \+ in the top left corner for " }{TEXT 262 9 "k = 1...n" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 167 "The leading_principal_minor_determina nts command computes and displays the determinants of the leading prin cipal minors and returns the sequence of these determinants." }}{PARA 15 "" 0 "" {TEXT -1 112 "In linear algebra: A matrix is positive defi nite if its leading principal minor determinants are all positive. " } }{PARA 15 "" 0 "" {TEXT -1 297 "In multivariable calculus: A critical \+ point p of a function f is a local minimum if the leading principal mi nor determinants of the Hessian matrix of f at p are all positive. It is a local maximum if the leading principal minor determinants of the Hessian alternate signs beginning with negative." }}{PARA 15 "" 0 "" {TEXT -1 383 "This command is part of the vec_calc package, and so can be used in the form leading_principal_minor_determinants only after p erforming the command with(vec_calc) or with(vec_calc, leading_princip al_minor_determinants). The command can always be accessed in the lon g form vec_calc[leading_principal_minor_determinants]. The alias LPMD can be used only after performing the command " }{HYPERLNK 17 "vc_ali ases" 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 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 -1 117 "- Copyright 1995-2001 by Ar thur 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[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 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } D! &(6}}{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#!$3sign simplif &    simplificat  simplify sin3   % & ' *  sinc singl sis   siv   sol solut solution solv some   spac %  spacecurv specif)  specificat#     % ' ( speed sph$   spher  spherical spiral sqrt  squar  &      ( namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      ss  step8   ) * still stor strip structur stud    submatric  submatrix subs subsequ substitut such   sumsuppos sure surfac+I % ) *    symbolic  symmetr system   tabl   tabulat tang    tangential    % & previou pricipal  principal&              % & ' ( ) *    % k{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 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 39 "The Jacobian matrix of \+ a function from " }{XPPEDIT 262 0 "R^k;" "6#)%\"RG%\"kG" }{TEXT 2 0 " " }{TEXT -1 4 " to " }{XPPEDIT 263 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 8 " is the " }{TEXT 261 5 "n x k" }{TEXT -1 118 " matrix whose ij'th e ntry is the partial derivatives of the i'th component function with re spect to the j'th variable." }}{PARA 15 "" 0 "" {TEXT -1 173 "JAC retu rns a list of lists of the first partial derivatives of the component \+ functions. The choice of variables is optional. To display the Jacob ian as an array, use the " }{HYPERLNK 17 "matrix" 2 "matrix" "" } {TEXT -1 9 " command." }}{PARA 15 "" 0 "" {TEXT -1 29 "JAC differs fro m 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 vector or list of expressions and return s a matrix of expressions. It requires the specification of the varia bles." }}{PARA 15 "" 0 "" {TEXT -1 226 "The function JAC is part of th e vec_calc package, and so can be used in the form JAC only after perf orming the command with(vec_calc) or with(vec_calc, JAC). The functio n can always be accessed in 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:=makefunction([u,v],[u^2+v^2,u+v,u*v]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG7%R6$%\"uG%\"vG6\"6$%)operatorG% &arrowGF*,&*$)9$\"\"#\"\"\"F3*$)9%F2F3F3F*F*F*RF'F*F+F*,&F1F3F6F3F*F*F *RF'F*F+F**&F1F3F6F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "JT:=JAC(T); matrix(JT(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #JTG7%7$R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF+,$9$\"\"#F+F+F+R6$F)F*F +F,F+,$9%F1F+F+F+7$\"\"\"F77$R6$F)F*F+F,F+F5F+F+F+R6$F)F*F+F,F+F0F+F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$,$%\"uG\"\"#,$%\" vGF*7$\"\"\"F.7$F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "T:= makefunction([rho,theta,phi], [rho*sin(phi)*cos(theta),\nrho*sin(phi)* sin(theta), rho*cos(phi)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG7 %R6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF+*(9$\"\"\"-%$sinG6#9 &F1-%$cosG6#9%F1F+F+F+RF'F+F,F+*(F0F1F2F1-F3F8F1F+F+F+RF'F+F,F+*&F0F1- F7F4F1F+F+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%R6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF,*&-%$sinG6 #9&\"\"\"-%$cosG6#9%F5F,F,F,R6%F)F*F+F,F-F,,$*(9$F5F1F5-F2F8F5!\"\"F,F ,F,R6%F)F*F+F,F-F,*(F>F5-F7F3F5F6F5F,F,F,7%R6%F)F*F+F,F-F,*&F1F5F?F5F, F,F,R6%F)F*F+F,F-F,*(F>F5F1F5F6F5F,F,F,R6%F)F*F+F,F-F,*(F>F5FDF5F?F5F, F,F,7%R6%F)F*F+F,F-F,FDF,F,F,\"\"!R6%F)F*F+F,F-F,,$*&F>F5F1F5F@F,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#R6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF(, $*&-%$sinG6#9&\"\"\")9$\"\"#F2!\"\"F(F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 " vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[jacob ian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops " 2 "Diffops" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "JAC_DET" 2 "JAC_DET" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "CoordConversion2D" 2 "CoordConversion2D" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "CoordConversion3D" 2 "CoordConversion3D" "" } {TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } F8F-F5F-7%F8\"\"!,$*&F4F-F)F-F6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JAC_DET(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%$rhoG%&thetaG%$phiG6\"6$%)operatorG%&arrowGF(, $*&-%$sinG6#9&\"\"\")9$\"\"#F2!\"\"F(F(F(" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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 "linalg[jacob ian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops " 2 "Diffops" "" }{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" "% functveccalcjaccalculatjacobianmatrixcoordinattransformatparametrsurfacspaccallsequencvarsparameterformvectorlistarrowdefinfunctionvariablnameusedindependdescriptwhosijthentrpartialderivatcomponwithrespectreturnfirstchoicoptionaldisplaarrausecommanddifferlinalgpackagactsexpressrequirspecificatpartcanonlyafterperformalwayaccesslongexamplmakefunctjtrhothetaphisincosdetcopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsodiffopmuintcoordconversdeg2raddotevall frenet jerk $leading_principal_minor_determinants lenlpmd makefunction normal p polar r  radrect s sph ss tangent tangential torsion vc_aliases vec_calc velocity & {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 260 20 "vec_calc[JAC_DET] - " }{TEXT -1 67 "Calculates the Ja cobian Determinant of a Coordinate Transformation " }}{PARA 0 "" 0 "" {TEXT 26 20 "Calling Sequences: \n" }{TEXT 256 43 " JAC_DET(T) \+ vec_calc[JAC_DET](T)" }}{PARA 0 "" 0 "" {TEXT 258 48 " JAC_DET(T ,vars) vec_calc[JAC_DET](T,vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "P arameters:" }{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 n variables " }}{PARA 0 "" 0 "" {TEXT 259 12 " vars - " }{TEXT -1 80 "a list of n names to be u sed as the independent variables of the transformation " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 98 "The Jacobian determinant of a coordinate transfor mation is the determinant of the Jacobian matrix." }}{PARA 15 "" 0 "" {TEXT -1 80 "JAC_DET returns an arrow-defined function. The choice of variables is optional." }}{PARA 15 "" 0 "" {TEXT -1 242 "The function JAC_DET is part of the vec_calc package, and so can be used in the fo rm JAC_DET only after performing the command with(vec_calc) or with(ve c_calc, JAC_DET). The function can always be accessed in the long for m vec_calc[JAC_DET]." }}}{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 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%R6%%$rhoG%&th etaG%$phiG6\"6$%)operatorG%&arrowGF+*(9$\"\"\"-%$sinG6#9&F1-%$cosG6#9% F1F+F+F+RF'F+F,F+*(F0F1F2F1-F3F8F1F+F+F+RF'F+F,F+*&F0F1-F7F4F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JT:=JAC(T);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%#JTG7%7%R6%%$rhoG%&thetaG%$phiG6\"6$%)operator G%&arrowGF,*&-%$sinG6#9&\"\"\"-%$cosG6#9%F5F,F,F,R6%F)F*F+F,F-F,,$*(9$ F5F1F5-F2F8F5!\"\"F,F,F,R6%F)F*F+F,F-F,*(F>F5-F7F3F5F6F5F,F,F,7%R6%F)F *F+F,F-F,*&F1F5F?F5F,F,F,R6%F)F*F+F,F-F,*(F>F5F1F5F6F5F,F,F,R6%F)F*F+F ,F-F,*(F>F5FDF5F?F5F,F,F,7%R6%F)F*F+F,F-F,FDF,F,F,\"\"!R6%F)F*F+F,F-F, ,$*&F>F5F1F5F@F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "JAC _DET(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%$rhoG%&thetaG%$phiG6\" 6$%)operatorG%&arrowGF(,$*&-%$sinG6#9&\"\"\")9$\"\"#F2!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "det(JT(rho,theta,phi)); simp lify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**()-%$sinG6#%$phiG\"\"$ \"\"\")-%$cosG6#%&thetaG\"\"#F+)%$rhoGF1F+!\"\"*(F%F+)-F'F/F1F+F2F+F4* *F2F+F&F+F6F+)-F.F(F1F+F4**F2F+F9F+F,F+F&F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#%$phiG\"\"\")%$rhoG\"\"#F)!\"\"" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur B elmonte 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 "JAC" 2 "JAC" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "linalg[j acobian]" 2 "linalg[jacobian]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "lina lg[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 "CoordConversion3D" 2 "CoordConver sion3D" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } 6#%&thetaG\"\"#F+)%$rhoGF1F+!\"\"*(F%F+)-F'F/F1F+F2F+F4* *F2F+F&F+F6F+)-F.F(F1F+F4**F2F+F9F+F,F+F&F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#%$phiG\"\"\")%$rhoG\"\"#F)!\"\"" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur B elmonte and Philipcommand{           % & ' ( ) *      complex  compon%  componentcomput+    ) * computat conclus confirm consist  constrain  constraint contain   ' (  contour  contourplot convers   convert     coordconvers   % &    velocit    vers         % & ' ( ) *     coordinat#9  % &    copyright{            % & ' ( ) *      corn  correspond cos/   % & ' * counterclockwis   critical   critpt cros  crossprodct   curl    ' (  curv+h !  *    spherical spiral sqrt  squar  &      ( namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      trig tripleint) true ' ( try type  unconstrain under  unfortunate unit  universit{            % & ' ( ) *      up   updat upper upward   use     %  usedwA           % & ' ( ) *      format % &  you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      derivat    % descript{            % & ' ( ) *      design      det  % & )  determin ' (  determinant*    &   df dfr diffdiffer'     % ' (  differential    differentiat   diffop/     % & ' (   & )  jacobian % &  ) *     partial  % path performo(          % & ' ( ) *      period permit *      matric matrix    % & max      ' {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 1 " \+ " }{TEXT 261 16 "vec_calc[POT] - " }{TEXT -1 81 "Calculates the Scalar 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 43 " POT(F,'f') vec_calc[POT](F,'f')" }} {PARA 0 "" 0 "" {TEXT 259 48 " POT(F,'f',vars) vec_calc[POT](F,' f',vars)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 12 " F - " }{TEXT -1 81 "a vector f ield in the form of a list of n arrow-defined functions of n variables " }}{PARA 0 "" 0 "" {TEXT 258 12 " 'f' - " }{TEXT -1 42 "the nam e for the potential to be returned " }}{PARA 0 "" 0 "" {TEXT 260 12 " \+ vars - " }{TEXT -1 57 "a list of n names to be used as the indepen dent variables" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 126 "POT determines whethe r a given vector field is the gradient of a scalar potential, and dete rmines that potential if it exists." }}{PARA 15 "" 0 "" {TEXT -1 88 "P OT returns true if the vector field F has a scalar potential, and fals e if it does not." }}{PARA 15 "" 0 "" {TEXT -1 150 "If a scalar potent ial for F exists, it will be assigned to the name given in the second \+ argument f. This second argument must be contained in quotes." }} {PARA 15 "" 0 "" {TEXT -1 29 "POT differs from the command " } {HYPERLNK 17 "potential" 2 "potential" "" }{TEXT -1 8 " in the " } {HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 161 " package:\nlinalg[p otential] acts on a vector or list of expressions and returns the pote ntial as an expression. It requires the specification of the variable s. " }}{PARA 15 "" 0 "" {TEXT -1 226 "The function POT is part of the \+ vec_calc package, and so can be used in the form POT only after perfor ming the command with(vec_calc) or with(vec_calc, POT). The function \+ can always be accessed in the long form vec_calc[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 44 "g:=makefunction([x,y,z],x^2+exp(y)*sin(z)); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF*,&*$)9$\"\"#\"\"\"F3*&-%$expG6#9%F3-%$sinG6#9&F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G:=GRAD(g); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arr owGF+,$9$\"\"#F+F+F+RF'F+F,F+*&-%$expG6#9%\"\"\"-%$sinG6#9&F8F+F+F+RF' F+F,F+*&F4F8-%$cosGF;F8F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "POT(G,'h'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(h); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF(,&*$)9$\" \"#\"\"\"F1*&-%$expG6#9%F1-%$sinG6#9&F1F1F(F(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 "ev al(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6%%\"aG%\"bG%\"cG6\"6$%)ope ratorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1*&-%$expG6#9%F1-%$sinG6#9&F1F1F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "F:=makefunction([x,y, z],[2*y, exp(y)*sin(z), exp(y)*cos(z)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,$9 %\"\"#F+F+F+RF'F+F,F+*&-%$expG6#F0\"\"\"-%$sinG6#9&F7F+F+F+RF'F+F,F+*& F4F7-%$cosGF:F7F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "PO T(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 "" {TEXT -1 117 "- Copyr ight 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Departm ent 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[potenti al]" "" }{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 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,$9 %\"\"#F+F+F+RF'F+F,F+*&-%$expG6#F0\"\"\"-%$sinG6#9&F7F+F+F+RF'F+F,F+*& F4F7-%$cosGF:F7F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "PO T(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 "" {TEXT -1 117 "- Copyr ight 1995-2001 by Arthur Belmonte and Philip B. Yasskin\n Departm ent of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_cuses using3     ) *   valu3*     ) *  var  * variablOO       % & ' ( ) *   vars+(     % & ' ( vcC,       ) *    vec{H       % & ' ( ) *      vecpot( vectorS    % & ' ( ) *    velocit    vers         % & ' ( ) *      period permit *      matric matrix    % & max      vertical   warren we well were   whatev wheth ' ( whil) * whos% will' ( with{n            % & ' ( ) *      without) * won work   written xn) yasskin{$            % & ' ( ) *      you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      ( {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {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%R6%%\"xG%\"yG%\"zG6\"6$%)o peratorG%&arrowGF+,(9$\"\"\"9%F19&F1F+F+F+RF'F+F,F+*(F0F1F2F1F3F1F+F+F +RF'F+F,F+,(*&F0F1F2F1F1*&F2F1F3F1F1*&F0F1F3F1F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C:=CURL(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,(9 $\"\"\"9&F1*&F0F19%F1!\"\"F+F+F+RF'F+F,F+,(F1F1F4F5F2F5F+F+F+RF'F+F,F+ ,&*&F4F1F2F1F1F1F5F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VEC_POT(C,'B');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF),*9&\"\" \"*&9%F/F.F/!\"\"*&#F/\"\"#F/*$)F.F5F/F/F2F1F/F)F)F)RF%F)F*F),(*&9$F/F .F/F2*&#F/F5F/F6F/F2*(F;F/F1F/F.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 "ev al(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%R6%%\"aG%\"bG%\"cG6\"6$%)o peratorG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*&#F/\"\"#F/*$)F.F5F/F/F2F1 F/F)F)F)RF%F)F*F),(*&9$F/F.F/F2*&#F/F5F/F6F/F2*(F;F/F1F/F.F/F/F)F)F)\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VEC_POT(A,'V');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"VG" }}} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur \+ Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 9 "See Also:" }{TEXT -1 1 " " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "linalg[vecpotent]" 2 "linalg[vecpotent]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Diffops" 2 "Diffops" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "CURL" 2 "CURL" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "DIV" 2 "DIV" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "POT" 2 "POT" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 }  ($ Siv rG%&arrowGF),*9&\"\"\"*&9%F/F.F/!\"\"*&#F/\"\"#F/*$)F.F5F/F/F2F1 F/F)F)F)RF%F)F*F),(*&9$F/F.F/F2*&#F/F5F/F6F/F2*(F;F/F1F/F.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 -1 117 "- Copyright 1995-2001 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" CURL CoordConversion2D CoordConversion3D Curve DIV Diffops GRAD HESS JAC% JAC_DET& LAP Line_int_scalar* Line_int_vectorMF Multi_Max_Min Multipleint) POT' Surface_int_scalarSurface_int_vectorVEC_POT( acceleration arclength binormal c ca can cat cb cforget cj ck cl cn crossct curvature curve curve_forget cv cyl ddegdeg2raddotevall frenet int * jerk $leading_principal_minor_determinants lenline* lis* livlpmd makefunction muint) multipleint) normal p polar r  radrect s scalar* sissivsph ss surfacetangent tangential torsion vc_aliases vec_calc vectorvelocity curve r          % & ' (  ! . 7 ; @ @ 11 x    @ x x   0 1b x    @ x x  ) w{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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\"\"$F.)%\"zG\"\"#F./F,;F.F4/F0;F1F-/F3;\"\"&\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "muint(x^4*y^3*z^2,x=1..2,y=3..4,z=5..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"&N()*\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "muint(x^4*y^3*z^2,x=1..2,y=3..4,z=5..6,`step`): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*()%\"xG\"\"%\" \"\")%\"yG\"\"$F.)%\"zG\"\"#F./F,;F.F4/F0;F1F-/F3;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$-F&6$*&-%'vectorG6#7#,$*()%\" xG\"\"&\"\"\")%\"yG\"\"$F4)%\"zG\"\"#F4#F4F3F4-%'matrixG6#7%7#&%!G6#/F 2F:7#FB7#&FB6#/F2F4F4/F6;F7\"\"%/F9;F3\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$-F&6$,$*&)%\"yG\"\"$\"\"\")%\"zG\"\"#F/# \"#J\"\"&/F-;F.\"\"%/F1;F5\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"~G-%$IntG6$*&-%'vectorG6#7#,$*&)%\"yG\"\"%\"\"\")%\"zG\"\"#F2#\"#J\" #?F2-%'matrixG6#7%7#&%!G6#/F0F17#F?7#&F?6#/F0\"\"$F2/F4;\"\"&\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$,$*$)%\"zG\"\"#\"\"\"# \"%&3\"\"\"%/F+;\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*& -%'vectorG6#7#,$*$)%\"zG\"\"$\"\"\"#\"%&3\"\"#7F/-%'matrixG6#7%7#&%!G6 #/F-\"\"'7#F97#&F96#/F-\"\"&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" ~G#\"&N()*\"#7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1 995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "Se e Also: " }{HYPERLNK 17 "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_s calar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int _vector" 2 "Line_int_vector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surfac e_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vector" "" }{TEXT -1 2 ". " }}} }{PAGENUMBERS 0 1 2 33 1 1 } " 0 "" {TEXT -1 117 "- Copyright 1 995-2001 by Arthur Belmonte and Philip B. Yasskin\n Department of Mathematics, Texas A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "Se e Also: " }{HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int " 2 "Int" "" }{TEXT -1 2 ", " }{HY incorrect independ/      % & ' (  indicat) * inert) input   insid instead  intu) *   integral$) *   integrand) integrat) * interior  intermediat) *  interpret into    introduct involv   iv   jac( % & )  jacobian % &  ) *     partial  % path performo(          % & ' ( ) *      period permit *      matric matrix    % & max      jame jared jerk   jt % & just ken lagrang lambda lap     laplacian   lead&      left   len  length lettletterlibnam limit) linalg;"      % & ' (  acobian % &    partial  % path perform_           % & ' (      period permit   left   len  length lettletterlibnam limit) linalg;"      % & ' (  assign' ( ) * assum at   availabl avoid   axes axis   back   backquot) * basical becom beginn belmont{!            % & ' ( ) *        %  array arrowK,         % & ' ( *  arthur{            % & ' ( ) *      namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      * &{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item " 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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#>% \"fGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,$*&9%\"\"\"9&F1#\"\"# \"\"*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r:=makefunct ion(t,[2*t,3*sin(t),3*cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG7%R6#%\"tG6\"6$%)operatorG%&arrowGF),$9$\"\"#F)F)F)RF'F)F*F),$-%$s inG6#F.\"\"$F)F)F)RF'F)F*F),$-%$cosGF4F5F)F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "Line_int_scalar(f,r,t=0..Pi/2); value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(-%$sinG6#%\"tG\"\"\"-%$co sGF*F,-%%sqrtG6#\"#8F,\"\"#/F+;\"\"!,$%#PiG#F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#8\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "line_int_scalar(f,r,t=0..Pi/2,`step`);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(-%$sinG6#%\"tG\"\"\"-%$cosGF*F,-%%s qrtG6#\"#8F,\"\"#/F+;\"\"!,$%#PiG#F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G*&-%'vectorG6#7#*&-%%sqrtG6#\"#8\"\"\")-%$sinG6#%\"tG\"\"# F/F/-%'matrixG6#7%7#&%!G6#/F4,$%#PiG#F/F57#F<7#&F<6#/F4\"\"!F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G,&*&-%%sqrtG6#\"#8\"\"\")-%$sinG 6#,$%#PiG#F+\"\"#F3F+F+*&F'F+)-F.6#\"\"!F3F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"#8\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=MF([x,y],3*x*sin(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\" \"-%$sinG6#9%F0\"\"$F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R:=MF(t,[ln(t),2*(-t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG 7$%#lnGR6#%\"tG6\"6$%)operatorG%&arrowGF*,$9$!\"#F*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#-%$IntG6$,$*&*(-%#lnG6#%\"tG\" \"\"-%$sinG6#,$F,\"\"#F--%%sqrtG6#,&*$)F,F2F-\"\"%F-F-F-F-F,!\"\"!\"$/ F,;F-F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$,$*&*(-%#lnG6#%\"t G\"\"\"-%$sinG6#,$F,\"\"#F--%%sqrtG6#,&*$)F,F2F-\"\"%F-F-F-F-F,!\"\"! \"$/F,;F-F2" }}}{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#>%\"hGR6&%\"xG%\"yG%\"zG%\"wG6\"6$%)operatorG%&arrowGF+ *(9$\"\"\"9&F19'F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:=MF(t,[2*t,t^2,t^2,2*t^3/3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG7&R6#%\"tG6\"6$%)operatorG%&arrowGF),$9$\"\"#F)F)F)RF'F)F*F)*$)F.F /\"\"\"F)F)F)RF'F)F*F)F1F)F)F)RF'F)F*F),$*$)F.\"\"$F3#F/F9F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "line_int_scalar(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#-%$IntG6$,$*&)%\"tG\"\"'\"\"\",&\"\"#F+*&F-F +)F)F-F+F+F+#\"\"%\"\"$/F);\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"~G*&-%'vectorG6#7#,&*$)%\"tG\"\"*\"\"\"#\"\")\"#F*&#F1\"#@F/)F-\" \"(F/F/F/-%'matrixG6#7%7#&%!G6#/F-F/7#F>7#&F>6#/F-\"\"!F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G#\"$G\"\"$*=" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_calc" 2 "v ec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makefunct ion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "int" 2 "int" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Line_int_vector" 2 "Line_int_vector" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vector" " " }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } 0 "6#/%\"~G#\"$G\"\"$*=" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthur Belmonte and Philip B. Y asskin\n Department of Mathematics, Texas A&M University " }} {PARA 0 "" 0 "" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "vec_calc" 2 "v * 8functionveccalclineintscalardisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandliscallsequencfcnvarrngparameterfunctvariablarrownotatcurvformlistdefinparametintegratrangoveroptionalindicatdescriptfirstargumsecondthirdevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignthespartpackagnameonlyperformalwayaccesslongexamplmakefunctsincospimflncopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintvectorsurfaccity V%{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item " 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "" 17 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 17 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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%R6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF+,$*&)9$\"\"#\"\"\"9%F4\"$V\"F+F+F+RF'F+F,F+,$ *&F5F49&F4$!$:(!\"\"F+F+F+RF'F+F,F+,$*&F2F4F:F4$\"#UF=F+F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r:=makefunction(t,[2*t^3,3*t ^4,t^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%R6#%\"tG6\"6$%)op eratorG%&arrowGF),$*$)9$\"\"$\"\"\"\"\"#F)F)F)RF'F)F*F),$*$)F0\"\"%F2F 1F)F)F)RF'F)F*F)*$)F0F3F2F)F)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\"\"!*&$\"%uDF.F+)F)\"\"*F+!\"\" *&$\"++++!o\"!\")F+)F)\"\"'F+F+/F);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 $R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF**$)9$\"\"$\"\"\"F*F*F*RF'F*F+F **$)F0\"\"&F2F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "R:=M F(t,[(cos(t))^3,(sin(t))^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"R G7$R6#%\"tG6\"6$%)operatorG%&arrowGF)*$)-%$cosG6#9$\"\"$\"\"\"F)F)F)RF 'F)F*F)*$)-%$sinGF1F3F4F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Liv(G,R,t=0..Pi/2); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$IntG6$,(*&)-%$cosG6#%\"tG\"#6\"\"\"-%$sinGF+F.!\"$*&\"\"$F.)F)\"#; F.F.*&F3F.)F)\"#=F.!\"\"/F,;\"\"!,$%#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&R6&%\"xG%\"yG%\"zG%\"wG6\"6$%) operatorG%&arrowGF,*$)9$\"\"#\"\"\"F,F,F,RF'F,F-F,*&F2F49'F4F,F,F,RF'F ,F-F,F7F,F,F,RF'F,F-F,*$)9&F3F4F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r:=MF(t,[sin(t),cos(t),sin(t),cos(t)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"rG7&%$sinG%$cosGF&F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "liv(H,r,t=0..Pi/2,'step');" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$,(*$)-%$cosG6#%\"tG\"\"#\"\"\"F.-%$sinGF+!\" \"*&F/F.F(F.F./F,;\"\"!,$%#PiG#F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"~G*&-%'vectorG6#7#,**&-%$cosG6#%\"tG\"\"\"-%$sinGF.F0#F0\"\"#*&F3 F0F/F0F0F,F0*&#F0\"\"$F0*$)F,F8F0F0!\"\"F0-%'matrixG6#7%7#&%!G6#/F/,$% #PiGF37#FB7#&FB6#/F/\"\"!F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G, 0*&-%$cosG6#,$%#PiG#\"\"\"\"\"#F--%$sinGF)F-F,*&#F-\"\"%F-F+F-F-F'F-*& #F-\"\"$F-*$)F'F6F-F-!\"\"*&#F-F.F-*&-F(6#\"\"!F--F0F>F-F-F9F=F9*&#F-F 6F-)F=F6F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#!\"#\"\"$\"\"\"*&#F '\"\"%F'%#PiGF'F'" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyrigh t 1995-2001 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 1 7 "Line _int_scalar" 2 "Line_int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Su rface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "Surface_int_vector" 2 "Surface_int_vector" "" }{TEXT -1 2 ". " }}}}{PAGENUMBERS 0 1 2 33 1 1 } \"\"\"*&#F '\"\"%F'%#PiGF'F'" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyrigh t 1995-2001 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 1 able abov   absolut  accelerat   accessg         % & ' ( ) *    acknowledgement action acts#     % ' ( advis after{5            % & ' ( ) *      algebra    algebraic alia      found fourth fr frenet   fsolv functsq        % & ' ( ) *        period permit     matric matrix    % & max      aliasCO       ) *    all+   ) *   allvalu also{"            % & ' ( ) *      alternat alwayg         % & ' ( ) *    analys analysi   angl   angular   % & ' ( ) *      format % &  you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max      ({VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 $} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 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 257 "" 0 "" {TEXT -1 11 " Sis = " }{TEXT 261 18 "Surface_int_scalar" }}{PARA 257 "" 0 "" {TEXT -1 11 " sis = " }{TEXT 264 18 "surface_int_scala r" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 262 18 "Surface_int_scalar " }{TEXT -1 45 "(fcn,R,var1=rng1,var2=rng2) Sis(...)" }} {PARA 256 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 263 18 "Surfa ce_int_scalar" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{TEXT 2&65 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 256 "" 0 "" {TEXT -1 3 " " }{TEXT 266 18 "surface_int_scalar" }{TEXT -1 52 "(fcn,R,var1=rng1,var2=rng2,`step`) sis(...,`step`)" }} {PARA 256 "" 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 49 "the variables of integration and curve parameters" }} {PARA 0 "" 0 "" {TEXT 270 17 " rng1,rng2 - " }{TEXT -1 46 "the ran ges of the parameters to integrate over" }}'{PARA 0 "" 0 "" {TEXT 260 17 " `step` - " }{TEXT -1 82 "an optional parameter to indicate that the intermediate steps should be displayed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 305 "Surface_int_scalar displays the surface integral of a sc alar field, where the first argument is the scalar field, the second a rgument is the surface and the third and fourth arguments are the vari ables of integration and their ranges. The integral can then be evalu ated using the value or evalf command. " }}{PARA 15 "" 0 "" {TEXT -1 108 "surface_int_scalar calculates the surface integral of a scalar fi eld without first displaying the integral. " }}{PARA 15 "" 0 "" {TEXT -1 258 "surface_int_scalar with the `step` parameter displays the surf ace 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 14 "The variables " }{TEXT 271 13 "var1 and var2" } {TEXT -1 208 " can appear in either order (whatever is best for the in tegration). However, the names of the variables of integration must m atch the names of the parameters used in the definition of the paramet ric surface." }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are par t of the vec_calc package, and so can be used by name only after perfo rming the command with(vec_calc) or with(vec_calc,function). The func tions can always be accessed in the long forms vec_calc[function]. Th e 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#>%\"fGR6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF*,&*&)9$\"\"#\"\"\"9%F3F3)*$)9&F2F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "R:=makefunction([theta,z],[3*cos(th eta),3*sin(theta),z]); # cylinder" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"RG7%R6$%&thetaG%\"zG6\"6$%)operatorG%&arrowGF*,$-%$cosG6#9$\"\"$F*F *F*RF'F*F+F*,$-%$sinGF1F3F*F*F*RF'F*F+F*9%F*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$6$,&*&)-%$co sG6#%&thetaG\"\"#\"\"\"-%$sinGF-F0\"#\")*&\"\"$F0)%\"zGF/F0F0/F.;\"\"! ,$%#PiGF//F7;F:F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Surface_int_scalar(f,R,z=0.. 2,theta=0..2*Pi); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$-F$6$,&*&)-%$cosG6#%&thetaG\"\"#\"\"\"-%$sinGF-F0\"#\")*&\"\"$F0)%\" zGF/F0F0/F7;\"\"!F//F.;F:,$%#PiGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$%#PiG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g:=MF([x,y,z ],sqrt(1+x^2+y^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"x*G% \"yG%\"zG6\"6$%)operatorG%&arrowGF**$-%%sqrtG6#,(\"\"\"F3*$)9$\"\"#F3F 3*$)9%F7F3F3F3F*F*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%R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF**&9$\"\" \"-%$cosG6#9%F0F*F*F*RF'F*F+F**&F/F0-%$sinGF3F0F*F*F*RF'F*F+F*F4F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "surface_int_scalar(g,S, u=0..1,v=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"%\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h:=MF([x,y,z],x*z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF**&9$\"\"\"9&F0F*F*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%R6$%&thetaG%$phiG6\"6$%)operatorG%&arrowGF*,$*&-%$sinG6#9%\" \"\"-%$cosG6#9$F4\"\"$F*F*F*RF'F*F+F*,$*&F0F4-F1F7F4F9F*F*F*RF'F*F+F*, $-F6F2F9F*F*F*" }}}{EXCHG {PARA 0 "+> " 0 "" {MPLTEXT 1 0 57 "surface_i nt_scalar(h,R,theta=0..Pi/2,phi=0..Pi/4,'step');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$**-%$sinG6#%$phiG\"\"\"-%$cosG6#%&theta GF.-F0F,F.-%%sqrtG6#,&F.F.*$)F3\"\"#F.!\"\"F.\"#\")/F2;\"\"!,$%#PiG#F. F:/F-;F?,$FA#F.\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6 $*&-%'vectorG6#7#,$**-%$sinG6#%$phiG\"\"\"-%$cosGF1F3-%%sqrtG6#,&F3F3* $)F4\"\"#F3!\"\"F3-F06#%&thetaGF3\"#\")F3-%'matrixG6#7%7#&%!G6#/F@,$%# PiG#F3F<7#FH7#&FH6#/F@\"\"!F3/F2;FS,$FL#F3\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$,&**-%$sinG6#%$phiG\"\"\"-%$cosGF,F.-%%s qrtG6#,&F.F.*$)F/\"\"#F.!\"\"F.-F+6#,$%#PiG#F.F7F.\"#\")*,F>F.F*F.F/F. F1F.-F+6#\"\"!F.F8/F-;FB,$F<#F.\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"~G*&-%'vectorG6#7#,$*$),&\"\"\"F.*$)-%$cosG6#%$phiG\"\"#F.!\"\"# \"\"$F5F.\"#FF.-%'matrixG6#7%7#&%!G6#/F4,$%#PiG#F.\"\"%7#F@7#&F@6#/F4 \"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G,&*$),&\"\"\"F)*$)-%$ cosG6#,$%#PiG#F)\"\"%\"\"#F)!\"\"#\"\"$F3F)\"#F*&F7F)),&F)F)*$)-F,-6#\" \"!F3F)F4F5F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"# \"\"\"#\"#F\"\"%" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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_vector" 2 "Surface_int_vector" "" }{TEXT -1 2 ". " }}}} {PAGENUMBERS 0 1 2 33 1 1 } -G6#\"\"# \"\"\"#\"#F\"\"%" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 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_vector" 2 "Surface_int_vector" "" }{TEXT -1 2 ". " }}}} {PAGENUMBERS 0 1 2 33 1 1 } -functionveccalcsurfacintscalardisplayintegralfieldcomputpossibintermediatstepaliascanusedafterexecutvccommandsiscallsequencfcnvarrngparameterfunctvariablarrownotatparametricformlistdefinintegratcurvrangoveroptionalparametindicatdescriptfirstargumsecondthirdfourthargumenttheirevaluatusingvaluevalfcalculatwithoutwithwhilallyoudoneedbackquotaroundassignappeareithorderwhatevbesthowevnamemustmatchdefinitthespartpackagonlyperformalwayaccesslongexamplmakefunctthetacossincylindpimfsqrtspiralrampphisphercopyrightarthurbelmontphilipyasskindepartmmathematictexauniversitalsomuintlinevector({VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0} {CSTYLE "" -1 266 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Cou rier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Fixed Width" -1 256 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 17 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 111 "Functions: " }{TEXT -1 1 "\n" }{TEXT 256 34 " vec_calc[Surface_int_vector] - " }{TEXT -1 46 " Displays a Surface Integral of a Vector Field " }}{PARA 0 "" 0 "" {TEXT 257 34 " vec_calc[surface_int_vector] - " }{TEXT -1 84 "Comput es a Surface Integral of a Vector 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 257 "" 0 "" {TEXT -1 11 " Siv = " }{TEXT 261 18 "Surface_int_vector" }}{PARA 257 "" 0 "" {TEXT -1 11 " siv = " }{TEXT 264 18 "surface_int_vecto r" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 262 18 "Surface_int_vector " }{TEXT -1 43 "(F,R,var1=rng1,var2=rng2) Siv(...)" }}{PARA 256 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 263 18 "Surface_int _vector" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {TEXT 265 128 "surface_int_vector" }{TEXT -1 43 "(F,R,var1=rng1,var2=rn g2) siv(...)" }}{PARA 256 "" 0 "" {TEXT -1 17 " vec_ca lc[" }{TEXT 267 18 "surface_int_vector" }{TEXT -1 6 "](...)" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 266 18 "surface_int_vector" } {TEXT -1 50 "(F,R,var1=rng1,var2=rng2,`step`) siv(...,`step`)" }} {PARA 256 "" 0 "" {TEXT -1 17 " vec_calc[" }{TEXT 268 18 "surfa ce_int_vector" }{TEXT -1 13 "](...,`step`)" }}{PARA 0 "" 0 "" {TEXT 26 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 258 17 " F - \+ " }{TEXT -1 85 "a vector field in the form of a list of 3 functions of 3 variables in arrow notation " }}{PARA 0 "" 0 "" {TEXT 259 17 " R \+ - " }{TEXT -1 89 "a parametric surface in the form of a list of 3 arrow-defined functions of two parameters" }}{PARA 0 "" 0 "" {TEXT 269 17 " var1,var2 - " }{TEXT -1 49 "the variables of integr ation and curve parameters" }}{PARA 0 "" 0 "" {TEXT 270 17 " rng1,rn g2 - " }{TEXT -1 46 "the ranges of the parame3ters to integrate over " }}{PARA 0 "" 0 "" {TEXT 260 17 " `step` - " }{TEXT -1 82 "an \+ optional parameter to indicate that the intermediate steps should be d isplayed." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" } {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 305 "Surface_int_vector dis plays the surface integral of a vector field, where the first argument is the vector field, the second argument is the surface and the third and fourth arguments are the variables of integration and their range s. The integral can then be evaluated using the value or evalf comman d. " }}{PARA 15 "" 0 "" {TEXT -1 108 "surface_int_vector calculates th e surface integral of a vector field without first displaying the inte gral. " }}{PARA 15 "" 0 "" {TEXT -1 259 "surface_int_vector with the ` step` parameter displays the surface integral of a vector field and th en calculates it while displaying all the intermediate steps. You do \+ not need the backquotes around `step` if the variable step has not bee n a4ssigned a value. " }}{PARA 15 "" 0 "" {TEXT -1 14 "The variables " }{TEXT 271 13 "var1 and var2" }{TEXT -1 208 " can appear in either ord er (whatever is best for the integration). However, the names of the \+ variables of integration must match the names of the parameters used i n the definition of the parametric surface." }}{PARA 15 "" 0 "" {TEXT -1 289 "These functions are part of the vec_calc package, and so can b e used by name only after performing the command with(vec_calc) or wit h(vec_calc,function). The functions can always be accessed in the lon g forms vec_calc[function]. The aliases can be used only after perfor ming 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_ca lc): vc_aliases:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F:=make function([x,y,z],[x*y,y+z^2,3*z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"FG7%R6%%\"xG%\"yG%\"zG6\"6$%)op5eratorG%&arrowGF+*&9$\"\"\"9%F1F+F+ F+RF'F+F,F+,&F2F1*$)9&\"\"#F1F1F+F+F+RF'F+F,F+,$F7\"\"$F+F+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%R6$%&thetaG%\"zG6\"6$%)operatorG%&arrowGF*,$-%$cosG6#9$ \"\"$F*F*F*RF'F*F+F*,$-%$sinGF1F3F*F*F*RF'F*F+F*9%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Surface_int_vector(F,R,theta=0..2*P i,z=0..2); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$ ,$*&-%$sinG6#%&thetaG\"\"\",(*$)-%$cosGF,\"\"#F.\"\"**&\"\"$F.F*F.F.*$ )%\"zGF4F.F.F.F7/F-;\"\"!,$%#PiGF4/F:;F=F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Surface_int_vector(F,R,z=0..2,theta=0..2*Pi); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$*&-%$sinG6#%&thetaG\"\"\",(* $)-%$cosGF,\"\"#F.\"\"**&\"\"$F.F*F.F.*$)%\"zGF4F.F.F.F7/F:;\"\"!F4/F- ;F=,$%#PiGF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#=" }}} {EXCHG {PARA6 0 "> " 0 "" {MPLTEXT 1 0 25 "G:=MF([x,y,z],[x*y,y,x]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%R6%%\"xG%\"yG%\"zG6\"6$%)oper atorG%&arrowGF+*&9$\"\"\"9%F1F+F+F+RF'F+F,F+F2F+F+F+RF'F+F,F+F0F+F+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% R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF**&9$\"\"\"-%$cosG6#9%F0F*F*F*RF 'F*F+F**&F/F0-%$sinGF3F0F*F*F*RF'F*F+F*F4F*F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "surface_int_vector(G,S,u=0..1,v=0..Pi,`step`); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$*(%\"uG\"\"\"-%$co sG6#%\"vGF+,(F*!\"#*&)F,\"\"#F+F*F+F+-%$sinGF.F+F+!\"\"/F*;\"\"!F+/F/; F:%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"~G-%$IntG6$*&-%'vectorG 6#7#,$*&-%$cosG6#%\"vG\"\"\",&*&,&!\"#F3*$)F/\"\"#F3F3F3)%\"uG\"\"$F3# F3F=*(#F3F:F3-%$sinGF1F3)F " 0 "" {MPLTEXT 1 0 29 "H:=MF([x,y,z],[y*z,x*z,x*y]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG7%R6%%\"xG%\"yG%\"zG6\"6$%) operatorG%&arrowGF+*&9%\"\"\"9&F1F+F+F+RF'F+F,F+*&9$F1F2F1F+F+F+RF'F+F ,F+*&F5F1F0F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "R:=M F([theta,phi], [3*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 3*cos(ph i)]); # sphere" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%R6$%&thetaG% $phiG6\"6$%)operatorG%&arrowGF*,$*&-%$sinG6#9%\"\"\"-%$cosG6#9$F4\"\"$ F*8F*F*RF'F*F+F*,$*&F0F4-F1F7F4F9F*F*F*RF'F*F+F*,$-F6F2F9F*F*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##!$V#\"#K" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthu r Belmonte and Philip B. Yasskin\n Department of Mathematics, Tex as A&M University " }}{PARA 0 "" 0 "" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "vec_calc" 2 "vec_calc" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "makefunction" 2 "makefunction" "" }{TEXT -1 2 ", " }{HYPERLNK 17 " Curve" 2 "Curve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "value" 2 "value" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Muint" 2 "Muint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int_scalar" 2 "Line_ int_scalar" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Line_int_vector" 2 "Lin e_int_vector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ". " 9}}}}{PAGENUMBERS 0 1 2 33 1 1 } {XPPMATH 20 "6##!$V#\"#K" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "- Copyright 1995-2001 by Arthu r Belmonte and Philip B. Yasskin\n Department of Mathematics, Tex as 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 "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 "Line_int_vector" 2 "Lin e_int_vector" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Surface_int_scalar" 2 "Surface_int_scalar" "" }{TEXT -1 2 ". " 9line?) *   linear lis *   listcN          % & ' ( *   listlist liv   ln *  load local   longg         % & ' ( ) *    lose lower lpmd     magnitudmultidimensional multipl)  multipleint)  multipli  multivariabl   must#   ' ( namec*          % & ' ( ) *   named need  ) * negat   nest neutral % & max      main make    makefunctS9        % & ' ( *    manipulat many mapl    maps match  mathematic{            % & ' ( ) *      matric matrix    % & max              % & ' ( ) *         you'   ) *  zero  namec*          % & ' ( ) *   named need  ) * negat   nest neutral  % & max       MathematicsMathematics/PackagesMathematics/Packages/vec_calc vec_calcvec_calcMathematics/Packages/vec_calcMathematics/Packages/vec_calc vec_calc