M7R0 I)vec_calc`6Xc&cross$:-1c%CURL$:-2c3curve_acceleration$:-3c0curve_arclength$:-4 c/curve_binormal$:-5c0curve_curvature$:-6c-curve_forget$:-7c+curve_jerk$:-8c-cu rve_normal$:-9c:curve_normal_acceleration%:-10c.curve_tangent%:-11c>curve_tange ntial_acceleration%:-12c.curve_torsion%:-13c/curve_velocity%:-14c)cyl2rect%:-15 c(cyl2sph%:-16c(deg2rad%:-17c$DIV%:-18c$dot%:-19c&evall%:-20c%GRAD%:-21c%HESS%: -22c$JAC%:-23c(JAC_DET%:-24c$LAP%:-25cEleading_principal_minor_determinants%:-2 6c$len%:-27c0line_int_scalar%:-28c0Line_int_scalar%:-29c0line_int_vector%:-30c0 Line_int_vector%:-31c-makefunction%:-32c,multipleint%:-33c,Multipleint%:-34c+po lar2rect%:-35c$POT%:-36c(rad2deg%:-37c)rect2cyl%:-38c+rect2polar%:-39c)rect2sph %:-40c(sph2cyl%:-41c)sph2rect%:-42c#ss%:-43c3surface_int_scalar%:-44c3Surface_i nt_scalar%:-45c3surface_int_vector%:-46c3Surface_int_vector%:-47c+vc_aliases%:- 48c(VEC_POT%:-49c,map_unapply%:-50c,MuInt_noChk%:-51c)Get_Vars%:-52c#&.%:-53c#& x%:-54b6#%+thismoduleG6"6$%(packageG%eqCopyright~1995-2001~by~Art~Belmonte~and~ Philip~B.~Yasskin,~Department~of~Mathematics,~Texas~A&M~UniversityG6X%&crossG%% CURLG%3curve_accelerationG%0curve_arclengthG%/curve_binormalG%0curve_curvatureG %-curve_forgetG%+curve_jerkG%-curve_normalG%:curve_normal_accelerationG%.curve_ tangentG%>curve_tangential_accelerationG%.curve_torsionG%/curve_velocityG%)cyl2 rectG%(cyl2sphG%(deg2radG%$DIVG%$dotG%&evallG%%GRADG%%HESSG%$JACG%(JAC_DETG%$LA PG%Eleading_principal_minor_determinantsG%$lenG%0line_int_scalarG%0Line_int_sca larG%0line_int_vectorG%0Line_int_vectorG%-makefunctionG%,multipleintG%,Multiple intG%+polar2rectG%$POTG%(rad2degG%)rect2cylG%+rect2polarG%)rect2sphG%(sph2cylG% )sph2rectG%#ssG%3surface_int_scalarG%3Surface_int_scalarG%3surface_int_vectorG% 3Surface_int_vectorG%+vc_aliasesG%(VEC_POTG%,map_unapplyG%,MuInt_noChkG%)Get_Va rsG%#&.G%#&xGFhn6#Qepa~collection~of~commands~which~simplify~calculations~arisi ng~from~vector~calculus~problemsFhnFhnFhn%,_m134716536G6"Fhn M7R0 I%:-27f*6#'%"vG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph ilip~B.~Yasskin,~Math,~TAMUGF*-%%sqrtG6$-%$addG6$*$&9$6#%"iG""#/F7;"""-%(vectdi mG6#F5%)symbolicGF*F*F*6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-17f*6#%&thetaG6"6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasski n,~Math,~TAMUGF&C$,$*&9$"""%#PiGF-#F-"$!=@%-%(hastypeG6$F,%&floatG-%&evalfG6#%" %GF9F&F&F&6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-48-%&aliasG6D/%#MFGc-makefunction%:-32/%#CvGc/curve_velocity%:-14/%#CaGc3cu rve_acceleration$:-3/%#CjGc+curve_jerk$:-8/%#CTGc.curve_tangent%:-11/%#CNGc-cur ve_normal$:-9/%#CBGc/curve_binormal$:-5/%#CkGc0curve_curvature$:-6/%#CtGc.curve _torsion%:-13/%#CLGc0curve_arclength$:-4/%$CaTGc>curve_tangential_acceleration% :-12/%$CaNGc:curve_normal_acceleration%:-10/%(CforgetGc-curve_forget$:-7/%$d2rG c(deg2rad%:-17/%$r2dGc(rad2deg%:-37/%$p2rGc+polar2rect%:-35/%$r2pGc+rect2polar% :-39/%$c2rGc)cyl2rect%:-15/%$r2cGc)rect2cyl%:-38/%$s2rGc)sph2rect%:-42/%$r2sGc) rect2sph%:-40/%$s2cGc(sph2cyl%:-41/%$c2sGc(cyl2sph%:-16/%&MuintGc,Multipleint%: -34/%&muintGc,multipleint%:-33/%%LPMDGcEleading_principal_minor_determinants%:- 26/%$LisGc0Line_int_scalar%:-29/%$lisGc0line_int_scalar%:-28/%$LivGc0Line_int_v ector%:-31/%$livGc0line_int_vector%:-30/%$SisGc3Surface_int_scalar%:-45/%$sisGc 3surface_int_scalar%:-44/%$SivGc3Surface_int_vector%:-47/%$sivGc3surface_int_ve ctor%:-466$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-38f*6#'%#rcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a ~list~or~vector~with~3~components.F*7%-T#6#&F36#;"""""#-%)simplifyG6#-%'arctanG 6$&F36#F?&F36#F>&F36#F4F*F*6$%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_ calcG M7R0 I%:-28f*6%%"fG'%"rG<$%'vectorG%%listG'%$rngG/%%nameG%&rangeG6%%"tG%%compG%&spee dG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$ -%$lhsG6#9&>8%-T#6#-9$6#-%#opG6#-9%6#F9>8&-FA6#-T%6#--%"DG6#FJFK@%/&9"6#!""%%st epG-FA6#-T'6%*&F?"""FMF^oF=Fgn-FA6#-F[o6$F]oF=F6F66(%#ssGc#ss%:-43%$lenGc$len%: -27%,multipleintGc,multipleint%:-336$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-18f*6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6&%%vars G%%nvarG%'varseqG%"iG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yassk in,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F=@$4-%%typeG6$F@-F)6#F.YQVThe~second~argu ment~must~be~a~list~of~variable~names.F8C$>F@-%&minusG6$<#-%$seqG6$-T#6#%"fG/FY 9$<#""!@'/-%%nopsG6#F@Fgn-%'RETURNG6#Fgn2"""FjnYQguThe~variables~were~different ~for~different~components|+~or~Maple's~automatic~simplification~changed~the~var iables.|+~You~may~specify~a~list~of~variables~as~a~second~argument.F8>F@-%#opGF \o>8%Fjn@$0-F[o6#FenFhoYQgnThe~number~of~components~must~equal~the~number~of~va riables.F8>8&Feo-%(unapplyG6$-%)simplifyG6#-%$addG6$--&%"DG6#8'6#&FenF^q6#F`p/F _q;FaoFhoF`pF8F86$%)Get_VarsGc)Get_Vars%:-526$%*protectedG/%+modulenameG%)vec_c alcG M7R0 I%:-49f*6$'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*%"fG6$%% varsG%&existG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math ,~TAMUG6"C%@%/9#""$C$>8$&9"6#F<@$4-%%typeG6$F?-F)6#F.YQUThe~third~argument~must ~be~a~list~of~variable~names.F7C$>F?-%&minusG6$<#-%$seqG6$-T#6#9%/FX9$<#""!@%0- %%nopsG6#F?"""YQgpCould~not~determine~the~variables.|+~You~may~specify~a~list~o f~variables~as~a~third~argument.F7>F?-%#opGF[o>8%-%*vecpotentG6%-FZ6#F`oF?.FX@% FcoC$>FX-T%6$F?-%(convertG6$-%%evalGFWF)%%trueG%&falseGF7F76&%)Get_VarsGc)Get_V ars%:-52%-makefunctionGc-makefunction%:-326$%*protectedG/%+modulenameG%)vec_cal cG M7R0 I%:-39f*6#'%#rcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""#YQYThe~argument~must~be~a ~list~or~vector~with~2~components.F*7$-T#F2-%)simplifyG6#-%'arctanG6$&F36#F4&F3 6#"""F*F*6$%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-29f*6%%"fG'%"rG<$%'vectorG%%listG'%$rngG/%%nameG%&rangeG6%%"tG%%compG%&spee dG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$ -%$lhsG6#9&>8%-T#6#-9$6#-%#opG6#-9%6#F9>8&-FA6#-T%6#--%"DG6#FJFK-T'6$*&F?"""FMF enF=F6F66(%#ssGc#ss%:-43%$lenGc$len%:-27%,MultipleintGc,Multipleint%:-346$%*pro tectedG/%+modulenameG%)vec_calcG M7R0 I%:-19f*6$'%"vG<$%'vectorG%%listG'%"wGF'6#%"nG6#%]oCopyright~1995-7~by~Art~Belm onte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%>8$-%(vectdimG6#9$@$0F3-F56#9%YQYThe ~two~vectors~must~have~the~same~number~of~components.F0-%)simplifyG6#-%$addG6$* &&F76#%"iG"""&F6$-F96#&F36#;FEFB&F36#F4F*F*6$%$lenGc$len%:-276$%*protectedG/%+mo dulenameG%)vec_calcG M7R0 I%:-30f*6%'%"FG<$%'vectorG%%listG'%"rGF''%$rngG/%%nameG%&rangeG6%%"tG%%compG%%v eloG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&> 8$-%$lhsG6#9&>8%-9$6#-%#opG6#-9%6#F:>8&--%"DG6#FHFI@%/&9"6#!""%%stepG-T%6#-T'6% -FX6#-T#6$F@FKF>FV-FX6#-Fen6$FgnF>F7F76(%$dotGc$dot%:-19%#ssGc#ss%:-43%,multipl eintGc,multipleint%:-336$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-20f*6#%"mG6"6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~M ath,~TAMUGF&-%(convertG6$-%&evalmG6#9$%)listlistGF&F&F&6$%*protectedG/%+modulen ameG%)vec_calcG M7R0 I%:-10f*6#'%"rG<$%'vectorG%%listG6&%"tG%"vG%"aG%"nG6$%)rememberG%^oCopyright~19 95,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F 7@$4-%%typeG6$F:%%nameGYQMThe~second~argument~must~be~a~variable~name.F2C$>F:-% &minusG6$<#-%$seqG6$-T#6#%"fG/FR9$<#""!@%0-%%nopsG6#F:"""C$-%'printfG6%Q'%s|+%s |+F2QLCould~not~determine~the~parameter.~Using~t.F2QRYou~may~specify~a~paramete r~as~a~second~argument.F2>F:.F:>F:-%#opG6#-FboFen>8%--T%6#FTFen>8&--T'FjoFen>8' -T+6#,&F\pFfn*(-T)6$F\pFfoFfn-Fhp6$FfoFfo!""FfoFfnF\q-%(unapplyG6$-%)simplifyG6 #*&-Fhp6$F\pFapFfn-T-6#FapF\qF:F2F26.%)Get_VarsGc)Get_Vars%:-52%/curve_velocity Gc/curve_velocity%:-14%3curve_accelerationGc3curve_acceleration$:-3%$dotGc$dot% :-19%&evallGc&evall%:-20%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_calcG M7R0 I$:-1f*6$'%"vG<$%'vectorG%%listG'%"wGF'6"6#%]oCopyright~1995-7~by~Art~Belmonte~ and~Philip~B.~Yasskin,~Math,~TAMUGF,C$@$50-%(vectdimG6#9$""$0-F46#9%F7YQZEach~a rgument~must~be~a~list~or~vector~with~3~components.F,7%,&*&&F66#""#"""&F;6#F7FD FD*&&F6FFFD&F;FBFD!"",&*&FHFD&F;6#FDFDFD*&&F6FNFDFEFDFJ,&*&FPFDFIFDFD*&FAFDFMFD FJF,F,F,6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-51f*6"6$%"iG%'my_intG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Y asskin,~Math,~TAMUGF$C$>8%&9"6#"""?(8$""#F09#%%trueG>F,-%$IntG6$F,&F.6#F2F$F$F$ 6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-41f*6#'%#scG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a ~list~or~vector~with~3~components.F*7%*&&F36#"""F;-%$sinG6#&F36#F4F;&F36#""#*&F 9F;-%$cosGF>F;F*F*F*6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I$:-2f*6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6#%%varsG 6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&@% /9#""#C$>8$&9"6#F:@$4-%%typeG6$F=-F)6#F.YQVThe~second~argument~must~be~a~list~o f~variable~names.F5C$>F=-%&minusG6$<#-%$seqG6$-T#6#%"fG/FV9$<#""!@'/-%%nopsG6#F =FZ-%'RETURNG6#7%FZFZFZ2"""FgnYQguThe~variables~were~different~for~different~co mponents|+~or~Maple's~automatic~simplification~changed~the~variables.|+~You~may ~specify~a~list~of~variables~as~a~second~argument.F5>F=-%#opGFin@$50-Fhn6#FX""$ 0FgnFjoYQLThere~must~be~3~components~and~3~variables.F5-7%,&-&%"DGF@6#&FX6#FjoF _o-&FcpFfp6#&FXF@!"",&-Fhp6#&FX6#F_oF_o-&FcpF`qFdpF[q,&-FbqFipF_o-FbpF^qF[q6#Fc o-T%6$F=-%)simplifyG6#%"%GF5F56&%)Get_VarsGc)Get_Vars%:-52%-makefunctionGc-make function%:-326$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-31f*6%'%"FG<$%'vectorG%%listG'%"rGF''%$rngG/%%nameG%&rangeG6%%"tG%%compG%%v eloG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&> 8$-%$lhsG6#9&>8%-9$6#-%#opG6#-9%6#F:>8&--%"DG6#FHFI-T'6$-T%6#-T#6$F@FKF>F7F76(% $dotGc$dot%:-19%#ssGc#ss%:-43%,MultipleintGc,Multipleint%:-346$%*protectedG/%+m odulenameG%)vec_calcG M7R0 I%:-21f*6#'%"fG%*procedureG6$%%varsG%"iG6#%^oCopyright~1995,96~by~Art~Belmonte~ and~Philip~B.~Yasskin,~Math,~TAMUG6"C$@%/9#""#C$>8$&9"6#F2@$4-%%typeG6$F5-%%lis tG6#%%nameGYQVThe~second~argument~must~be~a~list~of~variable~names.F-C$>F5-%&mi nusG6$<#-T#6#9$<#""!@%0-%%nopsG6#F5"""YQhpCould~not~determine~the~variables.|+~ You~may~specify~a~list~of~variables~as~a~second~argument.F->F5-%#opGFT-T%6$F5-% )simplifyG6#-7#-%$seqG6$-&%"DG6#8%FL/Fdo;FUFR6#FYF-F-6&%)Get_VarsGc)Get_Vars%:- 52%-makefunctionGc-makefunction%:-326$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-11f*6#'%"rG<$%'vectorG%%listG6$%"tG%"vG6$%)rememberG%^oCopyright~1995,96~by ~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%@%/9#""#C$>8$&9"6#F5@$4-%%t ypeG6$F8%%nameGYQMThe~second~argument~must~be~a~variable~name.F0C$>F8-%&minusG6 $<#-%$seqG6$-T#6#%"fG/FP9$<#""!@%0-%%nopsG6#F8"""C$-%'printfG6%Q'%s|+%s|+F0QLCo uld~not~determine~the~parameter.~Using~t.F0QRYou~may~specify~a~parameter~as~a~s econd~argument.F0>F8.F8>F8-%#opG6#-F`oFY>8%--T%6#FRFY-T+6$-%)simplifyG6#-T)6#*& FdoFZ-T'6#Fdo!""F8F0F06,%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc/curve_velo city%:-14%$lenGc$len%:-27%&evallGc&evall%:-20%,map_unapplyGc,map_unapply%:-506$ %*protectedG/%+modulenameG%)vec_calcG M7R0 I$:-3f*6#'%"rG<$%'vectorG%%listG6"6$%)rememberG%^oCopyright~1995,96~by~Art~Belm onte~and~Philip~B.~Yasskin,~Math,~TAMUGF*-%"DG6#-T#6#9$F*F*6$%/curve_velocityGc /curve_velocity%:-146$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-52f*6#%"fG6"6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math ,~TAMUGF&C$@$-%%typeG6$9$%*procedureG-%'RETURNG6#7#-%#opG6$"""-%%evalG6#F.""!F& F&F&6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-42f*6#'%#scG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a ~list~or~vector~with~3~components.F*7%*(&F36#"""F;-%$sinG6#&F36#F4F;-%$cosG6#&F 36#""#F;*(F9F;FF;F*F*F*6$%*protectedG/%+modulenameG%)vec_c alcG M7R0 I%:-32f*6$'%%varsG<$-%%listG6#%%nameGF+'%&exprsG<&-.%&arrayG6#%*algebraicG-F)F2 -%)listlistGF2F36"6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasski n,~Math,~TAMUGF7-T#6$9%-%#opG6#9$F7F76$%,map_unapplyGc,map_unapply%:-506$%*prot ectedG/%+modulenameG%)vec_calcG M7R0 I%:-22f*6#'%"fG%*procedureG6%%%nvarG%"iG%"jG6#%^oCopyright~1995,96~by~Art~Belmo nte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C$>8$-%%nopsG6#7#-%#opG6$"""-%%evalG6#9 $7#-%$seqG6$7#-F@6$-&%"DG6$8%8&F8$&9"6#F4@$4-%%typeG 6$F7%%nameGYQMThe~second~argument~must~be~a~variable~name.F/C$>F7-%&minusG6$<#- %$seqG6$-T#6#%"fG/FO9$<#""!@%0-%%nopsG6#F7"""C$-%'printfG6%Q'%s|+%s|+F/QLCould~ not~determine~the~parameter.~Using~t.F/QRYou~may~specify~a~parameter~as~a~secon d~argument.F/>F7.F7>F7-%#opG6#-F_oFX-%(unapplyG6$-%)simplifyG6#-%%diffG6$-T'6#- -T%6#FQFXF7F7F/F/6(%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc/curve_velocity% :-14%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_calcG M7R0 I$:-4f*6#'%"rG<$%'vectorG%%listG6%%"tG%"aG%"bG6$%)rememberG%^oCopyright~1995,96 ~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C$@%/9#""#C$>8$&9"6#F6@$4- %%typeG6$F9%%nameGYQMThe~second~argument~must~be~a~variable~name.F1C$>F9-%&minu sG6$<#-%$seqG6$-T#6#%"fG/FQ9$<#""!@%0-%%nopsG6#F9"""C$-%'printfG6%Q'%s|+%s|+F1Q LCould~not~determine~the~parameter.~Using~t.F1QRYou~may~specify~a~parameter~as~ a~second~argument.F1>F9.F9>F9-%#opG6#-FaoFZ-%(unapplyG6%-%$IntG6$-%)simplifyG6# -T'6#--T%6#FSFZ/F9;8%8&FfpFgpF1F16(%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc /curve_velocity%:-14%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-53f*6$'%"vG<$%'vectorG%%listG'%"wGF'6"6#%]oCopyright~1995-7~by~Art~Belmonte ~and~Philip~B.~Yasskin,~Math,~TAMUGF,-T#6$9$9%F,F,6$%$dotGc$dot%:-196$%*protect edG/%+modulenameG%)vec_calcG M7R0 I%:-43f*6#%%exprG6"6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin ,~Math,~TAMUGF&-%)simplifyG6$9$.%)symbolicGF&F&F&6$%*protectedG/%+modulenameG%) vec_calcG M7R0 I%:-33f*6"6+%*integrandG%"iG%*var_rangeG%$varG%%rngeG%(antiderG%&lowerG%&upperG %,subsantiderG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Mat h,~TAMUGF$C3@$29#""#YQXAn~integral~needs~at~least~one~variable~of~integration.F $@$0&9"6#!""%%stepG@%-%%typeG6$7#&F;6#;F5F4-%%listG6#/%%nameG%&rangeG-%'RETURNG 6#-%&valueG6#-T#6#F;YQUEach~variable~of~integration~must~be~a~name~=~range.F$@$ 4-FA6$7#&F;6#;F5!"#FGYFW-%&printG6#-FT6#&F;6#;"""Fjn>8$&F;6#Fdo?(8%F5Fdo,&F4Fdo FjnFdo%%trueGC,>8&&F;6#Fjo>8'-%$lhsG6#F_p>8(-%$rhsGFfp>8*-Fep6#Fhp>8+-FjpF^q>8) -%$intG6$FfoFcp>8,*&-%&arrayG6#7#FcqFdo-%'matrixG6#7%7#&%!G6#/FcpF`q7#Fdr7#&Fdr 6#/FcpF\qFdo>Ffo,&-%%subsG6$FfrFcqFdo-F_s6$F[sFcqF=-F]o6#/%"~G-FT6$Fhq&F;6#;,&F joFdoFdoFdo,&F4FdoF=Fdo-F]o6#/Ffs-FT6$FfoFis>F_p&F;6#Fjn>FcpFdp>FhpFip>F\qF]q>F `qFaq>FcqFdq>FhqFiq>FfoF]s-F]o6#/FfsFhq-F]o6#/FfsFfoFfoF$F$6$%,MuInt_noChkGc,Mu Int_noChk%:-516$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-23f*6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6%%%vars G%"iG%"jG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG 6"C$@%/9#""#C$>8$&9"6#F<@$4-%%typeG6$F?-F)6#F.YQVThe~second~argument~must~be~a~ list~of~variable~names.F7C$>F?-%&minusG6$<#-%$seqG6$-T#6#%"fG/FX9$<#""!@'/-%%no psG6#F?Ffn-%'RETURNG6#Ffn2"""FinYQguThe~variables~were~different~for~different~ components|+~or~Maple's~automatic~simplification~changed~the~variables.|+~You~m ay~specify~a~list~of~variables~as~a~second~argument.F7>F?-%#opGF[o7#-FS6$7#-FS6 $-&%"DG6#8%6#&FZ6#8&/F`p;F`oFin/Fdp;F`o-Fjn6#FZF7F76$%)Get_VarsGc)Get_Vars%:-52 6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-13f*6#'%"rG<$%'vectorG%%listG6$%"tG%#cpG6$%)rememberG%]oCopyright~1995-7~by ~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&@%/9#""#C$>8$&9"6#F5@$4-%%t ypeG6$F8%%nameGYQMThe~second~argument~must~be~a~variable~name.F0C$>F8-%&minusG6 $<#-%$seqG6$-T#6#%"fG/FP9$<#""!@%0-%%nopsG6#F8"""C$-%'printfG6%Q'%s|+%s|+F0QLCo uld~not~determine~the~parameter.~Using~t.F0QRYou~may~specify~a~parameter~as~a~s econd~argument.F0>F8.F8>F8-%#opG6#-F`oFY@$0-%(vectdimG6#FR""$YQMThe~torsion~is~ only~defined~in~3~dimensions.F0>8%-T)6$--T%FgoFY--T'FgoFY-%(unapplyG6$-%)simpli fyG6#*&-T-6$F\p--T+FgoFYFZ-F^q6$F\pF\p!""F8F0F06.%)Get_VarsGc)Get_Vars%:-52%/cu rve_velocityGc/curve_velocity%:-14%3curve_accelerationGc3curve_acceleration$:-3 %&crossGc&cross$:-1%+curve_jerkGc+curve_jerk$:-8%$dotGc$dot%:-196$%*protectedG/ %+modulenameG%)vec_calcG M7R0 I$:-5f*6#'%"rG<$%'vectorG%%listG6$%"tG%#cpG6$%)rememberG%]oCopyright~1995-7~by~ Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&@%/9#""#C$>8$&9"6#F5@$4-%%ty peG6$F8%%nameGYQMThe~second~argument~must~be~a~variable~name.F0C$>F8-%&minusG6$ <#-%$seqG6$-T#6#%"fG/FP9$<#""!@%0-%%nopsG6#F8"""C$-%'printfG6%Q'%s|+%s|+F0QLCou ld~not~determine~the~parameter.~Using~t.F0QRYou~may~specify~a~parameter~as~a~se cond~argument.F0>F8.F8>F8-%#opG6#-F`oFY@$0-%(vectdimG6#FR""$YQNThe~binormal~is~ only~defined~in~3~dimensions.F0>8%-T)6$--T%FgoFY--T'FgoFY-T/6$-%)simplifyG6#-T- 6#*&F\pFZ-T+6#F\p!""F8F0F060%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc/curve_ velocity%:-14%3curve_accelerationGc3curve_acceleration$:-3%&crossGc&cross$:-1%$ lenGc$len%:-27%&evallGc&evall%:-20%,map_unapplyGc,map_unapply%:-506$%*protected G/%+modulenameG%)vec_calcG M7R0 I%:-54f*6$'%"vG<$%'vectorG%%listG'%"wGF'6"6#%]oCopyright~1995-7~by~Art~Belmonte ~and~Philip~B.~Yasskin,~Math,~TAMUGF,-T#6$9$9%F,F,6$%&crossGc&cross$:-16$%*prot ectedG/%+modulenameG%)vec_calcG M7R0 I%:-44f*6&%"fG'%"rG<$%'vectorG%%listG'%*inner_rngG/%%nameG%&rangeG'%*outer_rngG F-6(%%compG%*lennormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmont e~and~Philip~B.~Yasskin,~Math,~TAMUG6"C)>8)-%#opG6$;"""""#-F@6$FC-%%evalG6#9%>8 (-FJ6#F>>8$-T#6#-9$6#-F@6#FL>8&-%%diffG6$FL&F>6#FC>8'-Ffn6$FL&F>6#FD>8%-FR6#-T' 6#-T%6$FZF[o@%/&9"6#!""%%stepG-FR6#-T)6&*&FPFCFaoFC9&9'F`p-FR6#-Fdp6%FfpFgpFhpF ;F;6*%#ssGc#ss%:-43%&crossGc&cross$:-1%$lenGc$len%:-27%,multipleintGc,multiplei nt%:-336$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-34f*6"F$6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasskin,~Mat h,~TAMUGF$C%@$29#""#YQXAn~integral~needs~at~least~one~variable~of~integration.F $@$4-%%typeG6$7#&9"6#;F+F*-%%listG6#<$%%nameG/F<%&rangeGYQinEach~variable~of~in tegration~must~be~a~name~or~a~name~=~range.F$-T#6#F5F$F$6$%,MuInt_noChkGc,MuInt _noChk%:-516$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-24f*6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6(%&ncom pG%%varsG%%nvarG%'varseqG%"iG%"jG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip ~B.~Yasskin,~Math,~TAMUG6"C)@%/9#""#C$>8%&9"6#F?@$4-%%typeG6$FB-F)6#F.YQVThe~se cond~argument~must~be~a~list~of~variable~names.F:C$>FB-%&minusG6$<#-%$seqG6$-T# 6#%"fG/Fen9$<#""!@'/-%%nopsG6#FBFin-%'RETURNG6#Fin2"""F\oYQguThe~variables~were ~different~for~different~components|+~or~Maple's~automatic~simplification~chang ed~the~variables.|+~You~may~specify~a~list~of~variables~as~a~second~argument.F: >FB-%#opGF^o>8$-F]o6#Fgn>8&F\o@$0FjoF^pYQgnThe~number~of~components~must~equal~ the~number~of~variables.F:>8'Fgo7#-FV6$7#-FV6$--&%"DG6#8(6#&Fgn6#8)6#Fdp/F`q;Fc oF^p/Fdq;FcoFjo-%(unapplyG6$-%)simplifyG6#-%$detG6#%"%GFdpF:F:6$%)Get_VarsGc)Ge t_Vars%:-526$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-14f*6#'%"rG<$%'vectorG%%listG6"6$%)rememberG%^oCopyright~1995,96~by~Art~Bel monte~and~Philip~B.~Yasskin,~Math,~TAMUGF*-%"DG6#9$F*F*F*6$%*protectedG/%+modul enameG%)vec_calcG M7R0 I$:-6f*6#'%"rG<$%'vectorG%%listG6&%"tG%"vG%"aG%"nG6$%)rememberG%^oCopyright~199 5,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F7 @$4-%%typeG6$F:%%nameGYQMThe~second~argument~must~be~a~variable~name.F2C$>F:-%& minusG6$<#-%$seqG6$-T#6#%"fG/FR9$<#""!@%0-%%nopsG6#F:"""C$-%'printfG6%Q'%s|+%s| +F2QLCould~not~determine~the~parameter.~Using~t.F2QRYou~may~specify~a~parameter ~as~a~second~argument.F2>F:.F:>F:-%#opG6#-FboFen>8%--T%6#FTFen>8&--T'FjoFen>8'- T+6#,&F\pFfn*(-T)6$F\pFfoFfn-Fhp6$FfoFfo!""FfoFfnF\q-%(unapplyG6$-%)simplifyG6# *(-Fhp6$F\pFapFfn-T-6#FapF\qFjpF\qF:F2F26.%)Get_VarsGc)Get_Vars%:-52%/curve_vel ocityGc/curve_velocity%:-14%3curve_accelerationGc3curve_acceleration$:-3%$dotGc $dot%:-19%&evallGc&evall%:-20%$lenGc$len%:-276$%*protectedG/%+modulenameG%)vec_ calcG M7R0 I%:-45f*6&%"fG'%"rG<$%'vectorG%%listG'%*inner_rngG/%%nameG%&rangeG'%*outer_rngG F-6(%%compG%*lennormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmont e~and~Philip~B.~Yasskin,~Math,~TAMUG6"C)>8)-%#opG6$;"""""#-F@6$FC-%%evalG6#9%>8 (-FJ6#F>>8$-T#6#-9$6#-F@6#FL>8&-%%diffG6$FL&F>6#FC>8'-Ffn6$FL&F>6#FD>8%-FR6#-T' 6#-T%6$FZF[o-T)6%*&FPFCFaoFC9&9'F;F;6*%#ssGc#ss%:-43%&crossGc&cross$:-1%$lenGc$ len%:-27%,MultipleintGc,Multipleint%:-346$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-35f*6#'%#pcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""#YQYThe~argument~must~be~a ~list~or~vector~with~2~components.F*7$*&&F36#"""F;-%$cosG6#&F36#F4F;*&F9F;-%$si nGF>F;F*F*F*6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-25f*6#'%'fprocsG<'%*algebraicG%)constantG%*procedureG%%listG%%nameG6$%'vars eqG%"iG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMU G6"@%-%%typeG6$9$<$%&arrayGF+-%$mapG6$T#F7C$>8$-%#opG6$"""-%%evalG6#F7-%(unappl yG6$-%)simplifyG6#-%$addG6$--&%"DG6$8%FVFG6#F@/FV;FD-%%nopsG6#7#F@F@F2F26$%$LAP Gc$LAP%:-256$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-15f*6#'%#ccG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a ~list~or~vector~with~3~components.F*7%*&&F36#"""F;-%$cosG6#&F36#""#F;*&F9F;-%$s inGF>F;&F36#F4F*F*F*6$%*protectedG/%+modulenameG%)vec_calcG M7R0 I$:-7f*6"F$6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math ,~TAMUGF$C--%'forgetG6$T#9"-F)6$T%F,-F)6$T'F,-F)6$T)F,-F)6$T+F,-F)6$T-F,-F)6$T/ F,-F)6$T1F,-F)6$T3F,-F)6$T5F,-F)6$T7F,F$F$68%/curve_velocityGc/curve_velocity%: -14%3curve_accelerationGc3curve_acceleration$:-3%+curve_jerkGc+curve_jerk$:-8%. curve_tangentGc.curve_tangent%:-11%-curve_normalGc-curve_normal$:-9%/curve_bino rmalGc/curve_binormal$:-5%0curve_curvatureGc0curve_curvature$:-6%.curve_torsion Gc.curve_torsion%:-13%0curve_arclengthGc0curve_arclength$:-4%>curve_tangential_ accelerationGc>curve_tangential_acceleration%:-12%:curve_normal_accelerationGc: curve_normal_acceleration%:-106$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-46f*6&'%"FG<$%'vectorG%%listG'%"rGF''%*inner_rngG/%%nameG%&rangeG'%*outer_r ngGF.6(%%compG%*ournormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belm onte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C)>8)-%#opG6$;"""""#-FA6$FD-%%evalG6#9 %>8(-FK6#F?>8$-9$6#-FA6#FM>8&-%%diffG6$FM&F?6#FD>8'-FZ6$FM&F?6#FE>8%-T#6$FXFin@ %/&9"6#!""%%stepG-T'6#-T)6&-F[p6#-T%6$FQF_o9&9'Fio-F[p6#-F^p6%F`pFepFfpF8$&9"6#F<@$4-%%typeG6$F?-F)6#F.YQUThe~third~argument~must ~be~a~list~of~variable~names.F7C$>F?-%&minusG6$<#-%$seqG6$-T#6#9%/FX9$<#""!@%0- %%nopsG6#F?"""YQgpCould~not~determine~the~variables.|+~You~may~specify~a~list~o f~variables~as~a~third~argument.F7>F?-%#opGF[o>8%-%*potentialG6%-FZ6#F`oF?.FX@% FcoC$>FX-T%6$F?-%%evalGFW%%trueG%&falseGF7F76&%)Get_VarsGc)Get_Vars%:-52%-makef unctionGc-makefunction%:-326$%*protectedG/%+modulenameG%)vec_calcG M7R0 I%:-26f*6#'%"HG<$%'matrixG%)listlistG6&%"nG%"kG%"DG%&LPMDsG6#%^oCopyright~1995, 96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C(>8$-%'rowdimG6#9$@$0F4 -%'coldimGF7YQ;The~matrix~must~be~square.F1>8'-%$seqG6$-%$detG6#-%*submatrixG6% F8;"""8%FJ/FL;FKF4-%'printfG6$Q$%s|+F1QFLeading~Principal~Minor~Determinants:F1 ?(FLFKFKF4%%trueG-%&printG6#/&8&6#FL&F@FfnF@F1F1F16$%*protectedG/%+modulenameG% )vec_calcG M7R0 I%:-16f*6#'%#ccG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~P hilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a ~list~or~vector~with~3~components.F*7%-T#6#7$&F36#"""&F36#F4&F36#""#-%)simplify G6#-%'arctanG6$F8$&9"6#F7 @$4-%%typeG6$F:%%nameGYQMThe~second~argument~must~be~a~variable~name.F2C$>F:-%& minusG6$<#-%$seqG6$-T#6#%"fG/FR9$<#""!@%0-%%nopsG6#F:"""C$-%'printfG6%Q'%s|+%s| +F2QLCould~not~determine~the~parameter.~Using~t.F2QRYou~may~specify~a~parameter ~as~a~second~argument.F2>F:.F:>F:-%#opG6#-FboFen>8%--T%6#FTFen>8&--T'FjoFen>8'- T+6#,&F\pFfn*(-T)6$F\pFfoFfn-Fhp6$FfoFfo!""FfoFfnF\q-T/6$-%)simplifyG6#-Fcp6#*& FapFfn-T-6#FapF\qF:F2F260%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc/curve_vel ocity%:-14%3curve_accelerationGc3curve_acceleration$:-3%$dotGc$dot%:-19%&evallG c&evall%:-20%$lenGc$len%:-27%,map_unapplyGc,map_unapply%:-506$%*protectedG/%+mo dulenameG%)vec_calcG M7R0 I%:-47f*6&'%"FG<$%'vectorG%%listG'%"rGF''%*inner_rngG/%%nameG%&rangeG'%*outer_r ngGF.6(%%compG%*ournormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belm onte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C)>8)-%#opG6$;"""""#-FA6$FD-%%evalG6#9 %>8(-FK6#F?>8$-9$6#-FA6#FM>8&-%%diffG6$FM&F?6#FD>8'-FZ6$FM&F?6#FE>8%-T#6$FXFin- T)6%-T'6#-T%6$FQF_o9&9'F