aŅ5 vec_calc/init2*‡–˜* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‡–˜)D with* student –˜* plots –˜* linalg readlib* mtaylorø–˜* piecewiseø–˜* polarø–˜* unassign: map_unapplyø–˜* vec_calc/map_unapply: MuInt_noChkø–˜* vec_calc/MuInt_noChk: type/alg_list6* list* alg_list array —˜ algebraic: type/proc_const_list6*—˜* proc_const_list—˜+—˜—˜ constant procedure name string: type/GDargs6*,+—˜,*,—˜6*—˜* integerZ—˜ø–˜* help/text/Curve: help/text/Cvb—˜: help/text/Cab—˜: help/text/Cjb—˜: help/text/CTb—˜: help/text/CNb—˜: help/text/CBb—˜: help/text/Ckb—˜: help/text/Ctb—˜: help/text/CLb—˜: help/text/CaTb—˜: help/text/CaNb—˜: help/text/Cforgetb—˜: help/text/Frenetb—˜ø–˜* help/text/map_Diff: help/text/map_Limit՗˜: help/text/map_Int՗˜ø–˜* help/text/muint: help/text/MuIntń—˜: vec_calc/Initialized true print* Initializing Package: vec_calc lprint* This package automatically loads the packages: ˜˜* student linalg plots˜˜* This package automatically loads the commands: ˜˜* mtaylor piecewise polar unassign˜˜* See the HELP on: ˜˜* vec_calc Curve Diffops CoordConversions˜˜* or any of the following commands:‡–˜*Ł–˜ģ–˜’–˜—˜I—˜i—˜q—˜y—˜—˜‘—˜‰—˜™—˜”—˜©—˜±—˜¹—˜Į—˜Ź—˜ų—˜Ż—˜ę—˜˜˜aŅ5 vec_calc/ss2* expr** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜ simplify*0 symbolic‹–˜‹–˜aŅ5 vec_calc/emcl2* m** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜ convert* evalm*0 list‹–˜‹–˜aŅ5 vec_calc/dot2* v w** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŽ–˜ dotprod*00 orthogonalŽ–˜Ž–˜aŅ5 vec_calc/norm22* v** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜ sqrt* dot*0¬–˜ symbolic‹–˜‹–˜aŅ5 vec_calc/cross2* v w** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŽ–˜ convert* crossprod*00 listŽ–˜Ž–˜aŅ5 vec_calc/map_Limit2* exprs var** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU–˜> type*06* array list map* map_Limit­–˜0 Limit*­–˜Ä–˜–˜–˜aŅ5 vec_calc/map_Diff2* exprs var** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU–˜> type*06* array list map* map_Diff­–˜0 Diff*­–˜Ä–˜–˜–˜aŅ5 vec_calc/map_Int2* exprs var** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU–˜> type*06* array list map* map_Int­–˜0 Int*­–˜Ć–˜–˜–˜aŅ5 vec_calc/map_unapply2* exprs* C* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*> type*06* array list map* map_unapply®–˜ 0’’’’*40D unapply*®–˜.ז˜Å–˜ subs*Ł–˜ "¦–˜¦–˜aŅ5 vec_calc/map_eval2* exprs** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU–˜> type*06* array list map* map_evalŖ–˜ eval*Ŗ–˜–˜–˜aŅ5 vec_calc/MF2*4 vars6* list* name”–˜4 exprs alg_list** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU£–˜ map_unapply*0 op*0£–˜£–˜aŅ5 vec_calc/GD2* fprocs* k*–˜D>& type*,*0’’’’ GDargs ERROR* invalid argument(s)>•–˜*06* array list map* GD³–˜ œ–˜*40D>Ī–˜:.Ō–˜:ז˜ op* œ–˜*Ģ–˜  D*ז˜*³–˜–˜–˜aŅ5 vec_calc/Cv2* r** remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŠ–˜ GD*0Š–˜Š–˜aŅ5 vec_calc/Ca2* r** remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŠ–˜ GD* Cv*0Š–˜Š–˜aŅ5 vec_calc/Cj2* r** remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŠ–˜ GD* Ca*0Š–˜Š–˜aŅ5 vec_calc/CT2* r* t v* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*°–˜*“–˜ map* eval0:. Cv*Ɩ˜*­–˜ map_unapply* simplify* emcl*ʖ˜“–˜ norm2*ʖ˜­–˜Ŗ–˜Ŗ–˜aŅ5 vec_calc/CB2* r* cp t* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*°–˜*“–˜ map* eval0:. crossprod* Cv*Ɩ˜*­–˜ CaŌ–˜Ö–˜ map_unapply* simplify* emcl*ʖ˜“–˜ norm2*ʖ˜­–˜Ŗ–˜Ŗ–˜aŅ5 vec_calc/CN2* r* t* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*­–˜*±–˜ map* eval0 crossprod* CB*Ą–˜*Ŗ–˜ CTĪ–˜Š–˜ map_unapply* simplify* convert* " listŖ–˜§–˜§–˜aŅ5 vec_calc/Ck2* r* t v* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*°–˜*“–˜ map* eval0:. Cv*Ɩ˜*­–˜ unapply* simplify* norm2* crossprod*ʖ˜ CaĶ–˜Ļ–˜“–˜ą–˜*ʖ˜­–˜Ŗ–˜Ŗ–˜aŅ5 vec_calc/Ct2* r* cp t* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*°–˜*“–˜ map* eval0:. crossprod* Cv*Ɩ˜*­–˜ CaŌ–˜Ö–˜ unapply* simplify* dot*ʖ˜ CjŌ–˜Ö–˜“–˜ī–˜*ʖ˜Ę–˜­–˜Ŗ–˜Ŗ–˜aŅ5 vec_calc/CL2* r* a b t* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*³–˜*·–˜ map* eval0 unapply* Int* simplify* norm2* Cv*ʖ˜*°–˜°–˜4..ģ–˜ī–˜­–˜­–˜aŅ5 vec_calc/CaT2* r* t* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*®–˜*²–˜ map* eval0 unapply* simplify* diff* norm2* Cv*Į–˜*«–˜«–˜«–˜Ø–˜Ø–˜aŅ5 vec_calc/CaN2* r* t v* remember Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*±–˜*µ–˜ map* eval0:. Cv*Ė˜*®–˜ unapply* simplify* CkĪ–˜Š–˜µ–˜ dot*Ē–˜Ē–˜µ–˜®–˜«–˜«–˜aŅ5 vec_calc/Cforget2*ˆ–˜* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUˆ–˜ D readlib* forget¦–˜* Cv0’’’’¦–˜* Ca°–˜¦–˜* Cj°–˜¦–˜* CT°–˜¦–˜* CB°–˜¦–˜* CN°–˜¦–˜* Ck°–˜¦–˜* Ct°–˜¦–˜* CL°–˜¦–˜* CaT°–˜¦–˜* CaN°–˜ˆ–˜ˆ–˜aŅ5 vec_calc/r2d2* theta** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŒ–˜D0 Pi“> hastype*„–˜ float evalf* "Ā–˜Œ–˜Œ–˜aŅ5 vec_calc/d2r2* theta** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMUŒ–˜D0 Pi§–˜§–˜“> hastype*„–˜ float evalf* "Ɩ˜Œ–˜Œ–˜aŅ5 vec_calc/p2r2* pc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* op*0©–˜ cos*„–˜*«–˜©–˜¤–˜©–˜ sin²–˜©–˜‹–˜‹–˜aŅ5 vec_calc/r2p2* rc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* norm2*,* op*40 simplify* arctan*¬–˜*³–˜µ–˜¬–˜*±–˜µ–˜‹–˜‹–˜aŅ5 vec_calc/c2r2* cc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* op*0©–˜ cos*„–˜*«–˜©–˜¤–˜©–˜ sin²–˜©–˜„–˜*«–˜‹–˜‹–˜aŅ5 vec_calc/r2c2* rc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* norm2*,* op*40 simplify* arctan*¬–˜*³–˜µ–˜¬–˜*±–˜µ–˜¬–˜*µ–˜‹–˜‹–˜aŅ5 vec_calc/s2r2* sc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* op*0©–˜ sin*„–˜*«–˜©–˜ cos*„–˜*«–˜©–˜¤–˜©–˜®–˜©–˜Ɩ˜¾–˜©–˜¤–˜©–˜»–˜²–˜©–˜‹–˜‹–˜aŅ5 vec_calc/r2s2* rc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* norm2*0 simplify* arctan* op*©–˜¹–˜*©–˜¬–˜*³–˜*¤–˜*,*¹–˜*4Ɩ˜½–˜©–˜¹–˜*©–˜‹–˜‹–˜aŅ5 vec_calc/c2s2* cc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* norm2*,* op*0¬–˜*²–˜¬–˜*²–˜ simplify* arctanŖ–˜‹–˜‹–˜aŅ5 vec_calc/s2c2* sc** Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‹–˜,* op*0©–˜ sin*„–˜*«–˜©–˜„–˜*«–˜¤–˜©–˜ cos²–˜©–˜‹–˜‹–˜aŅ5 vec_calc/MuInt_noChk2** i my_int* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‰–˜D:. 0’’’’*<.Ɩ˜0 true:©–˜ Int*©–˜ ¬–˜*²–˜‰–˜‰–˜aŅ5 vec_calc/MuInt2*‡–˜* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‡–˜D>"0 ERROR* An integral needs at least one variable of integration>& type*,* 0’’’’*4£–˜”–˜ list*6* name rangeՖ˜¦–˜* Each variable of integration must be a name or a name = range MuInt_noChk*ʖ˜‡–˜‡–˜aŅ5 vec_calc/muint2*‡–˜* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‡–˜ value* MuInt*0’’’’‡–˜‡–˜aŅ5 vec_calc/smuint2** integrand i var_range var rnge antiderivative* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU‡–˜D>"0 ERROR* An integral needs at least one variable of integration>& type*,* 0’’’’*4½–˜»–˜ list* name rangeĄ–˜* Each variable of integration must be a name = range print* MuInt_noChk*ą–˜:. ą–˜*<.½–˜—˜»–˜—˜—˜ trueD:. ą–˜*—˜:. lhs*.—˜:. rhs;—˜:. int*—˜5—˜:—˜ subs*5—˜A—˜*>—˜F—˜—˜S—˜*5—˜8—˜\—˜F—˜%—˜—˜* ,*—˜*F—˜ ą–˜*4—˜—˜—˜—˜»–˜.—˜—˜*n—˜—˜*—˜w—˜:.—˜ ą–˜*»–˜4—˜=—˜E—˜O—˜—˜*n—˜,*F—˜.—˜—˜*n—˜—˜—˜‡–˜*—˜aŅ5 vec_calc/GRAD2*4 f procedure* i* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*,* seq*  D*.*0·–˜4 nops*,* op*斘Ź–˜*斘ŗ–˜©–˜©–˜aŅ5 vec_calc/DIV2*4 F list*6* constant procedure name* ncomp varseq nvar i* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. nops*0:. op*ז˜*Ū–˜ map* evalі˜:.Ģ–˜*,*Ō–˜>ɖ˜ģ–˜ ERROR* The number of components must equal the number of variables. unapply* simplify* sum*  D*.* і˜)—˜ņ–˜*—˜4Ū–˜ģ–˜Ō–˜Ę–˜Ę–˜aŅ5 vec_calc/CURL2*4 F list*6* constant procedure name* ncomp varseq nvar* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. nops*0:. op*Ō–˜*Ų–˜ map* evalĪ–˜:.ɖ˜*,*і˜>(ʖ˜閘õ–˜ ERROR* There must be 3 components and 3 variables.,*  D** Ī–˜*õ–˜Ų–˜ —˜—˜* Ī–˜—˜ —˜* Ī–˜*Ų–˜Ų–˜ —˜/—˜—˜'—˜3—˜#—˜Ų–˜—˜,—˜'—˜ļ–˜ MF*ī–˜ simplify* "Ɩ˜Ć–˜aŅ5 vec_calc/LAP2*4 fprocs proc_const_list* varseq nvar i* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*> type*06* array list map* LAP»–˜D:. op*Ֆ˜*Ł–˜»–˜:. nops*,*Ņ–˜ unapply* simplify* sum*  D*.—˜*»–˜ź–˜—˜4Ł–˜į–˜Ņ–˜³–˜³–˜aŅ5 vec_calc/HESS2*4 f procedure* nvar i j* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. nops*,* op*¾–˜*Ā–˜0,* seq*,*Ķ–˜*  D*..*Ȗ˜Ž–˜4Ā–˜³–˜ܖ˜ä–˜°–˜°–˜aŅ5 vec_calc/LPMD2*4 H6* listlist matrix* k D LPMD* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. seq* det* submatrix*04.Ī–˜і˜4Ļ–˜ rowdim*Ģ–˜ print* LPMDs:<і˜Ļ–˜Ļ–˜Ų–˜ trueą–˜* .*і˜ ¹–˜ł–˜¹–˜¶–˜¶–˜aŅ5 vec_calc/JAC2*4 F list*6* constant procedure name* ncomp nvar i j* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. nops*0:.Ė–˜*,* op*Ū–˜*ߖ˜ map* evalŠ–˜,* seq*,*ņ–˜*  D*.* Š–˜*.—˜4ߖ˜Č–˜—˜4ߖ˜Ó–˜Å–˜Å–˜aŅ5 vec_calc/JAC_DET2*4 F list*6* constant procedure name* ncomp varseq nvar i j* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. nops*0:. op*Ū–˜*ߖ˜ map* evalՖ˜:.Š–˜*,*Ų–˜>Ķ–˜š–˜ ERROR* The number of components must equal the number of variables.,* seq*,*—˜*  D*.* Ֆ˜*.ö–˜-—˜4ߖ˜Ķ–˜'—˜4ߖ˜š–˜ unapply* simplify* det* "Ų–˜Ź–˜Ź–˜aŅ5 vec_calc/POT2*4 F6* vector*6* constant procedure name list“–˜ f* a consv* Copyright 1995 by Arthur Belmonte and Philip B. Yasskin, Math, TAMU*D:. op*Ļ–˜*Ӗ˜ map* eval0:. potential*ā–˜*Ģ–˜,š–˜0>喘D:õ–˜ MF*ņ–˜Ž–˜*õ–˜ true falseɖ˜É–˜aŅ5 help/text/vec_calc TEXT)* HELP FOR: Introduction to the vec_calc package  CALLING SEQUENCE: (args) vec_calc[](args)›–˜ SYNOPSIS: - The vec_calc package is a collection of functions designed to simplify calculations which arise from vector calculus problems. It is loaded by the command with(vec_calc); provided the system variable libname has been set to include the path to the package.›–˜ - The functions available are: CB CL CN CT CURL Ca CaN CaT Cforget Cj Ck Ct Cv DIV GD GRAD HESS JAC JAC_DET LAP LPMD MF MuInt POT c2r c2s cross d2r dot emcl map_Diff map_Int map_Limit map_eval muint norm2 p2r r2c r2d r2p r2s s2c s2r smuint ss›–˜ - As an example, to calculate the dot product of two vectors A and B, use›–˜ with(vec_calc,dot); dot(A,B);›–˜ - For more information on a particular function see vec_calc[].›–˜ - vec_calc package written by Arthur Belmonte and Philip B. Yasskin Package and Help Written by David Arnold and James Warren Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University with all rights reserved.›–˜ SEE ALSO: libname, with, vec_calc[], Curve, Diffops, CoordConversionsaŅ5 help/text/MF TEXT* FUNCTION: vec_calc[MF] - define a vector valued function  CALLING SEQUENCE: MF(f,g)–˜ PARAMETERS: f - a name or a list of names representing the independent variables g - an expression, vector of expressions, or a list of vector expressions–˜ EXAMPLE: > f:=MF([x,y,t],[[x^2,x+t,y-x^3*t^2],[t,y,x^2]]);–˜ 2 3 2 f := [[(x,y,t) -> x , (x,y,t) -> x + t, (x,y,t) -> y - x t ],–˜ 2 [(x,y,t) -> t, (x,y,t) -> y, (x,y,t) -> x ]]–˜ > f[1][2](xx,yy,tt);–˜ xx + tt–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calc, vec_calc[GD]aŅ5 help/text/Curve TEXT6* HELP FOR: vec_calc package, Frenet analysis of a curve. FUNCTIONS: vec_calc[Cv] - velocity vec_calc[Ca] - acceleration vec_calc[Cj] - jerk vec_calc[CT] - unit tangent vec_calc[CN] - unit normal vec_calc[CB] - unit binormal vec_calc[Ck] - curvature vec_calc[Ct] - torsion vec_calc[CL] - arc length vec_calc[CaT] - tangential acceleration vec_calc[CaN] - normal acceleration vec_calc[Cforget] - clear the remember tables–˜ CALLING SEQUENCE: Cv(r) Ca(r) Cj(r) CT(r) CN(r) CB(r) Ck(r) Ct(r) CL(r) CaT(r) CaN(r) Cforget(r)–˜ PARAMETERS: r - a curve in the form of a list of three procedures.–˜ SYNOPSIS: - Cx uses a remember table to speed up the computation. This table should be be cleared after finishing the Frenet analysis of a curve to avoid cluttering the memory. This is done by using the Cforget command.–˜ - These commands are designed to be used with the MF (Make Function) command in the vec_calc package and the spacecurve command in the plots package–˜ EXAMPLES: > r:=MF(t,[t*cos(t),t*sin(t),t]); > spacecurve([op(r(t)),t=-2*Pi..2*Pi]); > Cv(r); > Ca(r); > Cj(r); > CT(r); > CN(r); > CB(r); > Ck(r); > Ct(r); > L:=CL(r); L(0,2*Pi); value("); > CaT(r); > CaN(r); > Cforget(r);–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calc, vec_calc[MF], vec_calc[GD], spacecurveaŅ5 help/text/GD TEXT#* FUNCTION: vec_calc[GD] - generalized derivative command for vector calculus  CALLING SEQUENCE: GD(F,G)”–˜ PARAMETERS: F - a procedure, constant, name or a list thereof G - a list of integers representing the variables that are differentiated with respect to.”–˜ SYNOPSIS: - GD returns an error if the type sent in is *, such as -sin.”–˜ EXAMPLES: > g:=(x,y) -> x^2+x^3*y;”–˜ 2 3 g := (x,y) -> x + x y > GD(g,[1,2]);”–˜ 2 (x,y) -> 3 x”–˜ > GD(g);”–˜ 2 (x,y) -> 2 x + 3 x y”–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University”–˜ SEE ALSO: vec_calc, vec_calc[Cv], vec_calc[Ca], vec_calc[Cj], vec_calc[CT], vec_calc[CB], vec_calc[CN], vec_calc[Ck], vec_calc[Ct], vec_calc[CL], vec_calc[CaT], vec_calc[CaN]aŅ5 help/text/DIV TEXT** FUNCTION: vec_calc[DIV] - computes the procedural divergence  CALLING SEQUENCE: DIV([f1,...,fn])ž–˜ PARAMETERS: f1 - a procedure in n variables f2 ... fn - each a procedure in n variables, name or a constantž–˜ SYNOPSIS: - DIV will not work if the first component of the inputed list is not a procedure of n arguments, n being the number of items in the list.ž–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4; g:=(x,y,z)->y^2*z; h:=(x,y,z)->x^2*z^3;ž–˜ 2 3 4 f := (x,y,z) -> x + y + x y zž–˜ 2 g := (x,y,z) -> y zž–˜ 2 3 h := (x,y,z) -> x zž–˜ > F:=[f,g,h]; map(eval,F);ž–˜ F := [f, g, h]ž–˜ 2 3 4 2 2 3 [(x,y,z) -> x + y + x y z , (x,y,z) -> y z, (x,y,z) -> x z ]ž–˜ > DIV(F);ž–˜ 4 2 2 (x,y,z) -> 2 x + y z + 2 y z + 3 x zž–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M Universityž–˜ SEE ALSO: vec_calc, vec_calc[CURL]aŅ5 help/text/CURL TEXT** FUNCTION: vec_calc[CURL] - determines the procedural curl of a three dimensional vector field  CALLING SEQUENCE: CURL([f1,f2,f3])©–˜ PARAMETERS: [f1,f2,f3] - a list of three procedures in three variables©–˜ SYNOPSIS: - CURL will not work if the first procedure f1 is a constant or built-in function©–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4; g:=(x,y,z)->y^2*z; h:=(x,y,z)->x^2*z^3;©–˜ 2 3 4 f := (x,y,z) -> x + y + x y z©–˜ 2 g := (x,y,z) -> y z©–˜ 2 3 h := (x,y,z) -> x z©–˜ > F:=[f,g,h]; map_eval(F);©–˜ F := [f, g, h]©–˜ 2 3 4 2 2 3 [(x,y,z) -> x + y + x y z , (x,y,z) -> y z, (x,y,z) -> x z ]©–˜ > C := CURL(F);©–˜ 2 3 3 2 4 C := [(x,y,z) -> - y , (x,y,z) -> 4 x y z - 2 x z , (x,y,z) -> - 3 y - x z ]©–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University©–˜ SEE ALSO: vec_calc, vec_calc[Diffops], vec_calc[DIV]aŅ5 help/text/LAP TEXT* FUNCTION: vec_calc[LAP] - computes the procedural Laplacian  CALLING SEQUENCE: LAP(fproc)–˜ PARAMETERS: fproc - a list of procedures, constants, names or list of combinations–˜ SYNOPSIS: - LAP will not work if the first component is not a procedure with the number of arguments equal to the size of the list.–˜ - LAP finds the laplacian in n-dimensions, where n is the number of arguments in the first componet of the inputed list.–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4;–˜ 2 3 4 f := (x,y,z) -> x + y + x y z > LAP(f);–˜ 2 (x,y,z) -> 2 + 6 y + 12 x y z–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calc, vec_calc[HESS]aŅ5 help/text/HESS TEXT* FUNCTION: vec_calc[HESS] - computes the procedural Hessian  CALLING SEQUENCE: HESS(proc)–˜ PARAMETERS: proc - a procedure–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4;–˜ 2 3 4 f := (x,y,z) -> x + y + x y z–˜ > K:=HESS(f);–˜ 4 3 K := [[2, (x,y,z) -> z , (x,y,z) -> 4 y z ],–˜ 4 3 [(x,y,z) -> z , (x,y,z) -> 6 y, (x,y,z) -> 4 x z ],–˜ 3 3 2 [(x,y,z) -> 4 y z , (x,y,z) -> 4 x z , (x,y,z) -> 12 x y z ]]–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calc, vec_calc[CURL], vec_calc[DIV], vec_calc[JAC]aŅ5 help/text/LPMD TEXT* FUNCTION: vec_calc[LPMD] - Leading Principal Minor Determinants  CALLING SEQUENCE: LPMD(fproc)ž–˜ PARAMETERS: fproc - a square matrixž–˜ SYNOPSIS: - LPMD returns the sequence of the determinants of the successively larger square matrices down the principal diagonal of the given matrixž–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M Universityž–˜ SEE ALSO: vec_calcaŅ5 help/text/JAC TEXT.* FUNCTION: vec_calc[JAC] - computes the procedural Jacobian  CALLING SEQUENCE: JAC(list)–˜ PARAMETERS: list - a list of procedures, constants, or names, or combinations thereof–˜ SYNOPSIS: - JAC will not work if the first component is a constant.–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4; g:=(x,y,z)->y^2*z; h:=(x,y,z)->x^2*z^3;–˜ 2 3 4 f := (x,y,z) -> x + y + x y z–˜ 2 g := (x,y,z) -> y z–˜ 2 3 h := (x,y,z) -> x z–˜ > F:=[f,g,h]; map(eval,F);–˜ F := [f, g, h]–˜ 2 3 4 2 2 3 [(x,y,z) -> x + y + x y z , (x,y,z) -> y z, (x,y,z) -> x z ]–˜ > J:=JAC(F);–˜ 4 3 J := [[(x,y,z) -> 2 x + y z , 0, (x,y,z) -> 2 x z ],–˜ 2 4 [(x,y,z) -> 3 y + x z , (x,y,z) -> 2 y z, 0],–˜ 3 2 2 2 [(x,y,z) -> 4 x y z , (x,y,z) -> y , (x,y,z) -> 3 x z ]]–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calc, vec_calc[JAC_DET], vec_calc[HESS]aŅ5 help/text/JAC_DET TEXT** FUNCTION: vec_calc[JAC_DET] - computes the procedural Jacobian determinant  CALLING SEQUENCE: JAC_DET(f1..fn)¢–˜ PARAMETERS: f1 - a procedure of n variables f2,..,fn - a list of procedures, constants, or names¢–˜ SYNOPSIS: - JAC_DET will not work if the first component is not a procedure of n variables where n is the number of componets in the list sent in.¢–˜ EXAMPLE: > f:=(x,y,z)->x^2+y^3+x*y*z^4; g:=(x,y,z)->y^2*z; h:=(x,y,z)->x^2*z^3;¢–˜ 2 3 4 f := (x,y,z) -> x + y + x y z¢–˜ 2 g := (x,y,z) -> y z¢–˜ 2 3 h := (x,y,z) -> x z¢–˜ > F:=[f,g,h]; map_eval(F);¢–˜ F := [f, g, h]¢–˜ 2 3 4 2 2 3 [(x,y,z) -> x + y + x y z , (x,y,z) -> y z, (x,y,z) -> x z ]¢–˜ > JAC_DET(F);¢–˜ 3 3 2 2 7 3 4 (x,y,z) -> 12 y z x - 8 x y z + 6 x z y¢–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University¢–˜ SEE ALSO: vec_calc, vec_calc[JAC], vec_calc[HESS]aŅ5 help/text/POT TEXT* FUNCTION: vec_calc[POT] - computes the procedural Potential  CALLING SEQUENCE: POT(G,f)–˜ PARAMETERS: G - either a list or vector of procedures, constants, or names f - name for the potential to be returned as.–˜ SYNOPSIS: - POT will be false unless G is conservative. - POT will not work unless the first element of G contains the same number of arguments as there are componets in G.–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University–˜ SEE ALSO: vec_calcaŅ5 help/text/dot TEXT * FUNCTION: vec_calc[dot] - compute the dot product of two vectors  CALLING SEQUENCE: dot(a, b)Ÿ–˜ PARAMETERS: a, b - lists or vectors of the same lengthŸ–˜ SYNOPSIS: - vec_calc[dot] is a modification of linalg[dotprod] where the 'orthogonal' option is always selected.Ÿ–˜ - This function is part of the vec_calc package, and so can be used in the form dot(..) only after performing the command with(linalg) or with(vec_calc,dot). The function can always be accessed in the long form vec_calc[dot](..).Ÿ–˜ EXAMPLE: > with(vec_calc): > a:=[1,2,3]; b:=[4,5,6]; dot(a,b);Ÿ–˜ a := [1, 2, 3]Ÿ–˜ b := [4, 5, 6]Ÿ–˜ 32Ÿ–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M UniversityŸ–˜ SEE ALSO: linalg[dotprod], vec_calc, vec_calc[cross]aŅ5 help/text/map_Diff TEXTD* HELP FOR: vec_calc package, mapping operations onto vector valued functions  FUNCTIONS: vec_calc[map_Limit] - determines the limit on a list vec_calc[map_Int] - performs integration on a list vec_calc[map_Diff] - performs differenation of a list£–˜ CALLING SEQUENCE: map_Limit(r,t=value) map_Diff(r,t) map_Int(r,t)£–˜ PARAMETERS: r - a list of procedures, constants or names or a list thereof. t - a variable representing the independent variable. value - number that t approaches in map_Limit procedure.£–˜ EXAMPLES: > r := MF(t,[(t^2-4)/(t-2),(t^2-5*t+6)/(t-2)]);£–˜ 2 2 t - 4 t - 5 t + 6 r := [t -> ------, t -> ------------] t - 2 t - 2£–˜ > map_Limit(r(t),t=2);£–˜ 2 2 t - 4 t - 5 t + 6 [Limit ------, Limit ------------] t -> 2 t - 2 t -> 2 t - 2£–˜ > value(");£–˜ [4, -1]£–˜ > map_Diff([(t^2-4)/(t-2),(t^2-5*t+6)/(t-2)],t);£–˜ 2 2 d t - 4 d t - 5 t + 6 [---- ------, ---- ------------] dt t - 2 dt t - 2£–˜Ų—˜£–˜ 2 2 t t - 4 2 t - 5 t - 5 t + 6 [2 ----- - --------, ------- - ------------] t - 2 2 t - 2 2 (t - 2) (t - 2)£–˜ > map_Int(r(t),t);£–˜ / 2 / 2 | t - 4 | t - 5 t + 6 [ | ------ dt, | ------------ dt] | t - 2 | t - 2 / /£–˜Ų—˜£–˜ 2 2 [1/2 t + 2 t, 1/2 t - 3 t]£–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University£–˜ SEE ALSO: vec_calc, vec_calc[MF], Limit, Diff, IntaŅ5 help/text/d2r TEXT* FUNCTION: vec_calc[d2r] - converts degrees to radians  CALLING SEQUENCE: d2r(theta)œ–˜ PARAMETERS: theta - a variable or set of variablesœ–˜ SYNOPSIS: - d2r returns a float if a float is entered. If theta is symbolic then a symbolic answer is returned.œ–˜ EXAMPLE: > d2r(a);œ–˜ 1/180 a Piœ–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M Universityœ–˜ SEE ALSO: vec_calc, vec_calc[r2d]aŅ5 help/text/r2d TEXT* FUNCTION: r2d - converts radians to degrees  CALLING SEQUENCE: r2d(theta)™–˜ PARAMETERS: theta - a variable or set of variables™–˜ SYNOPSIS: - r2d returns a float if a float is entered. If theta is symbolic then a symbolic answer is returned.™–˜ EXAMPLE: > r2d(a);™–˜ a 180 ---- Pi™–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University™–˜ SEE ALSO: vec_calc, vec_calc[d2r]aŅ5 help/text/p2r TEXT* FUNCTION: vec_calc[p2r] - converts polar coordinates to rectangular coordinates  CALLING SEQUENCE: p2r(r,theta)¢–˜ PARAMETERS: r - a variable representing the radius theta - a variable representing the angle in radians.¢–˜ SYNOPSIS: - p2r returns a coordinate pair of floats (x,y) if r or theta is a float.¢–˜ EXAMPLE: > p2r([a,b]);¢–˜ [a cos(b), a sin(b)]¢–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University¢–˜ SEE ALSO: vec_calc, vec_calc[r2p]aŅ5 help/text/r2p TEXT* FUNCTION: r2p - converts rectangular coordinates to polar coordinates  CALLING SEQUENCE: r2p(x,y) –˜ PARAMETERS: x - a variable y - a variable –˜ SYNOPSIS: - r2p will return a pair of coordinates (r,theta). r is always a nonnegative number and theta is between -Pi and Pi. –˜ EXAMPLE: > r2p([a,b]); –˜ 2 2 1/2 [(a + b ) , arctan(b, a)] –˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University –˜ SEE ALSO: vec_calc, vec_calc[p2r]aŅ5 help/text/c2r TEXT* FUNCTION: vec_calc[c2r] - converts cylindrical coordinates to rectangular coordinates  CALLING SEQUENCE: c2r(r,theta,z)§–˜ PARAMETERS: r - a variable representing the radius theta - a variable in radians z - a variable representing the z axis§–˜ SYNOPSIS: - c2r will return the ordered pair (x, y, z) where x = r * cos(theta), y = r * sin(theta), and z = z. If either r or theta are floats then x and y will be floats.§–˜ EXAMPLE: > c2r([a,b,c]);§–˜ [a cos(b), a sin(b), c]§–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University§–˜ SEE ALSO: vec_calc, vec_calc[r2c]aŅ5 help/text/r2c TEXT* FUNCTION: vec_calc[r2c] - converts rectangular coordinates to cylindrical coordinates  CALLING SEQUENCE: r2c(x,y,z)§–˜ PARAMETERS: x - a variable y - a variable z - a variable§–˜ SYNOPSIS: - r2c will return the ordered pair (r, thera, z) where r^2 = x^2 + y^2, theta = arctan(y/x), z = z. r is always a nonnegative number and theta is between -Pi and Pi§–˜ EXAMPLE: > r2c([a,b,c]);§–˜ 2 2 1/2 [(a + b ) , arctan(b, a), c]§–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University§–˜ SEE ALSO: vec_calc, vec_calc[c2r]aŅ5 help/text/s2r TEXT* FUNCTION: s2r - converts spherical coordinates to rectangular coordinates  CALLING SEQUENCE: s2r(rho,phi,theta)”–˜ PARAMETERS: rho - a variable representing the radius phi - a variable representing the aximuthal angle theta - a variable in radians representing angle in the x,y plane”–˜ SYNOPSIS: - s2r will return the ordered triplet (x, y, z) where x = rho * cos(theta) * sin(phi), y = rho * sin(theta) * sin(phi), z = rho * cos(phi).”–˜ - s2r assumes a right handed coordinate system”–˜ EXAMPLE: > s2r([a,b,c]);”–˜ [a sin(b) cos(c), a sin(b) sin(c), a cos(b)]”–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University”–˜ SEE ALSO: vec_calc, vec_calc[r2s]aŅ5 help/text/r2s TEXT* FUNCTION: vec_calc[r2s] - converts rectangular coordinates to spherical coordinates  CALLING SEQUENCE: r2s(x,y,z)¦–˜ PARAMETERS: x - a variable y - a variable z - a variable¦–˜ SYNOPSIS: - r2s returns the ordered triplet (rho, theta, phi) where rho is nonnegative and both theta and phi are between -Pi..Pi.¦–˜ - r2s will return a right handed coordinate system.¦–˜ EXAMPLE: > r2s([a,b,c]);¦–˜ 2 2 2 1/2 2 2 1/2 [(a + b + c ) , arctan((a + b ) , c), arctan(b, a)]¦–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University¦–˜ SEE ALSO: vec_calc, vec_calc[s2r]aŅ5 help/text/c2s TEXT* FUNCTION: vec_calc[c2s] - converts cylindrical coordinates to spherical coordinates  CALLING SEQUENCE: c2s(r,theta,z)¦–˜ PARAMETERS: r - a variable representing the radius theta - a variable in radians representing angle in the x,y plane z - a variable representing the height along the z-axis¦–˜ SYNOPSIS: - c2s will return the ordered pair (rho, theta, phi) where phi is between -Pi and Pi - c2s returns a right handed coordinate system¦–˜ EXAMPLE: > c2s([a,b,c]);¦–˜ 2 2 1/2 [(a + c ) , arctan(a, c), b]¦–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University¦–˜ SEE ALSO: vec_calc, vec_calc[s2c]aŅ5 help/text/s2c TEXT* FUNCTION: vec_calc[s2c] - converts spherical coordinates to cylindrical coordinates  CALLING SEQUENCE: s2c(rho,phi,theta)¦–˜ PARAMETERS: rho - a variable representing the radius phi - a variable representing the azimuthal angle theta - a variable in radians representing angle in the x,y plane¦–˜ SYNOPSIS: - s2c will return the ordered triplet (r,theta,z) - s2c returns assumes a right handed coordinate system¦–˜ EXAMPLE: > s2c([a,b,c]);¦–˜ [a sin(b), c, a cos(b)]¦–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University¦–˜ SEE ALSO: vec_calc, vec_calc[c2s]aŅ5 help/text/muint TEXT0* FUNCTIONS: vec_calc[MuInt] - Display Inert Mulitiple Integral vec_calc[muint] - Evaluates Multiple Integral  CALLING SEQUENCE: MuInt(F,x1,x2,..,x_n) muint(F,x1,x2,..,x_n)Ɩ˜ PARAMETERS: F - a procedure, the integrand x1 ... x_n - each must be a name or a range, the variables of integration in order of evaluationƖ˜ SYNOPSIS: - MuInt will not work if there are not at least two argumentsƖ˜ EXAMPLES: > MuInt(x*y*z,x,y,z);Ɩ˜ / / / | | | | | | x y z dx dy dz3—˜3—˜ / / /Ɩ˜ > MuInt(x^2*y^3*z^4,x=1..2,y=1..2,z=1..2);Ɩ˜ 2 2 2 / / / | | | 2 3 4 | | | x y z dx dy dz | | | / / / 1 1 1Ɩ˜ > muint(sin(x),x=2..y,y=1..z,z=2..3);Ɩ˜ cos(3) + sin(1) + 1/2 cos(2)Ɩ˜ > muint(x^2*y^3*z^4,x=1..2,y=1..2,z=1..2);Ɩ˜ 217/4Ɩ˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M UniversityƖ˜ SEE ALSO: vec_calc, intaŅ5 help/text/smuint TEXT:* FUNCTION smuint - Stepwise Multiple Integral  CALLING SEQUENCE: smuint(F,x1,x2,..,xn)›–˜ PARAMETERS: F - an expression to be used as the integrand x1 ... xn - each must be a name = range, the variables of integration in order of evaluation›–˜ SYNOPSIS: - smuint will not work if there are not at least two arguments›–˜ EXAMPLE: > smuint(x^2*y^3*z^4,x=1..2,y=1..2,z=1..2);›–˜ 2 2 2 / / / | | | 2 3 4 | | | x y z dx dy dz | | | / / / 1 1 1›–˜ 3 3 4 Anti-Derivative = [1/3 x y z , x = 1 .. 2]›–˜ 2 2 / / | | 3 4 = | | 7/3 y z dy dz | | / / 1 1›–˜ 4 4 Anti-Derivative = [7/12 y z , y = 1 .. 2]›–˜ 2 / | 4 = | 35/4 z dz | / 1›–˜ 5 Anti-Derivative = [7/4 z , z = 1 .. 2]›–˜ = 217/4›–˜ 217/4›–˜ - Copyright 1995 by Arthur Belmonte and Philip B. Yasskin Department of Mathematics, Texas A&M University›–˜ SEE ALSO: vec_calc, vec_calc[muint]aŅ5 vec_calc8* falseJH CT readlib* vec_calc/CTH CB“–˜* vec_calc/CBH Ck“–˜* vec_calc/CkH Ct“–˜* vec_calc/CtH CaT“–˜* vec_calc/CaTH CaN“–˜* vec_calc/CaNH CL“–˜* vec_calc/CLH CN“–˜* vec_calc/CNH r2d“–˜* vec_calc/r2dH d2r“–˜* vec_calc/d2rH map_Limit“–˜* vec_calc/map_Limit p2r“–˜* vec_calc/p2rH map_Diff“–˜* vec_calc/map_DiffH map_eval“–˜* vec_calc/map_evalH r2p“–˜* vec_calc/r2pH c2r“–˜* vec_calc/c2rH r2c“–˜* vec_calc/r2cH s2r“–˜* vec_calc/s2rH r2s“–˜* vec_calc/r2sH c2s“–˜* vec_calc/c2sH s2c“–˜* vec_calc/s2cH DIV“–˜* vec_calc/DIVH ss“–˜* vec_calc/ssH LAP“–˜* vec_calc/LAPH emcl“–˜* vec_calc/emcl dot“–˜* vec_calc/dotH norm2“–˜* vec_calc/norm2H cross“–˜* vec_calc/crossH map_Int“–˜* vec_calc/map_IntH Cforget“–˜* vec_calc/CforgetH MuInt“–˜* vec_calc/MuIntH JAC“–˜* vec_calc/JACH muint“–˜* vec_calc/muintH smuint“–˜* vec_calc/smuintH GRAD“–˜* vec_calc/GRADH CURL“–˜* vec_calc/CURLH HESS“–˜* vec_calc/HESS POT“–˜* vec_calc/POTH LPMD“–˜* vec_calc/LPMDH JAC_DET“–˜* vec_calc/JAC_DETH MF“–˜* vec_calc/MFH GD“–˜* vec_calc/GDH Cv“–˜* vec_calc/CvH Ca“–˜* vec_calc/CaH Cj“–˜* vec_calc/Cj