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I%:-40R6#'%#rcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
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I%:-41R6#'%#scG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
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G6#%%nameGYQVThe~second~argument~must~be~a~list~of~variable~names.6"C$>F5-%&min
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peG6$F8%%nameGYQMThe~second~argument~must~be~a~variable~name.6"C$>F8-%&minusG6$
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I%:-42R6#'%#scG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
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$F7%%nameGYQMThe~second~argument~must~be~a~variable~name.6"C$>F7-%&minusG6$<#-%
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I%:-13R6#'%"rG<$%'vectorG%%listG6$%"tG%#cpG6$%)rememberG%]oCopyright~1995-7~by~
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yG6#*&-T-6$F]p--T+FhoFZFen-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#/%+modulename
G%)vec_calcG
M6R0
I$:-6R6#'%"rG<$%'vectorG%%listG6&%"tG%"vG%"aG%"nG6$%)rememberG%^oCopyright~1995
,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F7@
$4-%%typeG6$F:%%nameGYQMThe~second~argument~must~be~a~variable~name.6"C$>F:-%&m
inusG6$<#-%$seqG6$-T#6#%"fG/FS9$<#""!@%0-%%nopsG6#F:"""C$-%'printfG6%Q'%s|+%s|+
FFQLCould~not~determine~the~parameter.~Using~t.FFQRYou~may~specify~a~parameter~
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cityGc/curve_velocity%:-14%3curve_accelerationGc3curve_acceleration$:-3%$dotGc$
dot%:-19%&evallGc&evall%:-20%$lenGc$len%:-276#/%+modulenameG%)vec_calcG
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I%:-54R6$'%"vG<$%'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$:-16#/%+modu
lenameG%)vec_calcG
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I%:-44R6&%"fG'%"rG<$%'vectorG%%listG'%*inner_rngG/%%nameG%&rangeG'%*outer_rngGF
-6(%%compG%*lennormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmonte
~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,multiplein
t%:-336#/%+modulenameG%)vec_calcG
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I%:-34R6"F$6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math
,~TAMUGF$C%@$29#""#YQXAn~integral~needs~at~least~one~variable~of~integration.6"
@$4-%%typeG6$7#&9"6#;F+F*-%%listG6#<$%%nameG/F=%&rangeGYQinEach~variable~of~int
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noChk%:-516#/%+modulenameG%)vec_calcG
M6R0
I%:-24R6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6(%&ncomp
G%%varsG%%nvarG%'varseqG%"iG%"jG6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~
B.~Yasskin,~Math,~TAMUG6"C)@%/9#""#C$>8%&9"6#F?@$4-%%typeG6$FB-F)6#F.YQVThe~sec
ond~argument~must~be~a~list~of~variable~names.6"C$>FB-%&minusG6$<#-%$seqG6$-T#6
#%"fG/Ffn9$<#""!@'/-%%nopsG6#FBFjn-%'RETURNG6#Fjn2"""F]oYQguThe~variables~were~
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d~the~variables.|+~You~may~specify~a~list~of~variables~as~a~second~argument.FO>
FB-%#opGF_o>8$-F^o6#Fhn>8&F]o@$0F[pF_pYQgnThe~number~of~components~must~equal~t
he~number~of~variables.FO>8'Fho7#-FW6$7#-FW6$--&%"DG6#8(6#&Fhn6#8)6#Fep/Faq;Fdo
F_p/Feq;FdoF[p-%(unapplyG6$-%)simplifyG6#-%$detG6#%"%GFepF:F:6$%)Get_VarsGc)Get
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M6R0
I%:-14R6#'%"rG<$%'vectorG%%listG6"6$%)rememberG%^oCopyright~1995,96~by~Art~Belm
onte~and~Philip~B.~Yasskin,~Math,~TAMUGF*-%"DG6#9$F*F*F*6#/%+modulenameG%)vec_c
alcG
M6R0
I$:-7R6"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$T/F
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14%3curve_accelerationGc3curve_acceleration$:-3%+curve_jerkGc+curve_jerk$:-8%.c
urve_tangentGc.curve_tangent%:-11%-curve_normalGc-curve_normal$:-9%/curve_binor
malGc/curve_binormal$:-5%0curve_curvatureGc0curve_curvature$:-6%.curve_torsionG
c.curve_torsion%:-13%0curve_arclengthGc0curve_arclength$:-4%>curve_tangential_a
ccelerationGc>curve_tangential_acceleration%:-12%:curve_normal_accelerationGc:c
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M6R0
I%:-45R6&%"fG'%"rG<$%'vectorG%%listG'%*inner_rngG/%%nameG%&rangeG'%*outer_rngGF
-6(%%compG%*lennormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmonte
~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$l
en%:-27%,MultipleintGc,Multipleint%:-346#/%+modulenameG%)vec_calcG
M6R0
I%:-35R6#'%#pcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
ilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""#YQYThe~argument~must~be~a~
list~or~vector~with~2~components.6"7$*&&F36#"""F<-%$cosG6#&F36#F4F<*&F:F<-%$sin
GF?F<F*F*F*6#/%+modulenameG%)vec_calcG
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I%:-25R6#'%'fprocsG<'%*algebraicG%)constantG%*procedureG%%listG%%nameG6$%'varse
qG%"iG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG
6"@%-%%typeG6$9$<$%&arrayGF+-%$mapG6$T#F7C$>8$-%#opG6$"""-%%evalG6#F7-%(unapply
G6$-%)simplifyG6#-%$addG6$--&%"DG6$8%FVFG6#F@/FV;FD-%%nopsG6#7#F@F@F2F26$%$LAPG
c$LAP%:-256#/%+modulenameG%)vec_calcG
M6R0
I%:-15R6#'%#ccG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
ilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a~
list~or~vector~with~3~components.6"7%*&&F36#"""F<-%$cosG6#&F36#""#F<*&F:F<-%$si
nGF?F<&F36#F4F*F*F*6#/%+modulenameG%)vec_calcG
M6R0
I$:-8R6#'%"rG<$%'vectorG%%listG6"6$%)rememberG%^oCopyright~1995,96~by~Art~Belmo
nte~and~Philip~B.~Yasskin,~Math,~TAMUGF*-%"DG6#-T#6#9$F*F*6$%3curve_acceleratio
nGc3curve_acceleration$:-36#/%+modulenameG%)vec_calcG
M6R0
I%:-46R6&'%"FG<$%'vectorG%%listG'%"rGF''%*inner_rngG/%%nameG%&rangeG'%*outer_rn
gGF.6(%%compG%*ournormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmo
nte~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$FQF_o9&9'Fio-F[p6#-F^p6%F`pFepFfpF<F<6*%&c
rossGc&cross$:-1%$dotGc$dot%:-19%#ssGc#ss%:-43%,multipleintGc,multipleint%:-336
#/%+modulenameG%)vec_calcG
M6R0
I%:-36R6$'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*%"fG6$%%v
arsG%&existG6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,
~TAMUG6"C%@%/9#""$C$>8$&9"6#F<@$4-%%typeG6$F?-F)6#F.YQUThe~third~argument~must~
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%nopsG6#F?"""YQgpCould~not~determine~the~variables.|+~You~may~specify~a~list~of
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FdoC$>FY-T%6$F?-%%evalGFX%%trueG%&falseGF7F76&%)Get_VarsGc)Get_Vars%:-52%-makef
unctionGc-makefunction%:-326#/%+modulenameG%)vec_calcG
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I%:-26R6#'%"HG<$%'matrixG%)listlistG6&%"nG%"kG%"DG%&LPMDsG6#%^oCopyright~1995,9
6~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C(>8$-%'rowdimG6#9$@$0F4-
%'coldimGF7YQ;The~matrix~must~be~square.6">8'-%$seqG6$-%$detG6#-%*submatrixG6%F
8;"""8%FK/FM;FLF4-%'printfG6$Q$%s|+F?QFLeading~Principal~Minor~Determinants:F??
(FMFLFLF4%%trueG-%&printG6#/&8&6#FM&FAFgnFAF1F1F16#/%+modulenameG%)vec_calcG
M6R0
I%:-16R6#'%#ccG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
ilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a~
list~or~vector~with~3~components.6"7%-T#6#7$&F36#"""&F36#F4&F36#""#-%)simplifyG
6#-%'arctanG6$F=F@F*F*6$%$lenGc$len%:-276#/%+modulenameG%)vec_calcG
M6R0
I$:-9R6#'%"rG<$%'vectorG%%listG6&%"tG%"vG%"aG%"nG6$%)rememberG%^oCopyright~1995
,96~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F7@
$4-%%typeG6$F:%%nameGYQMThe~second~argument~must~be~a~variable~name.6"C$>F:-%&m
inusG6$<#-%$seqG6$-T#6#%"fG/FS9$<#""!@%0-%%nopsG6#F:"""C$-%'printfG6%Q'%s|+%s|+
FFQLCould~not~determine~the~parameter.~Using~t.FFQRYou~may~specify~a~parameter~
as~a~second~argument.FF>F:.F:>F:-%#opG6#-FcoFfn>8%--T%6#FUFfn>8&--T'F[pFfn>8'-T
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bpFgn-T-6#FbpF]qF:F2F260%)Get_VarsGc)Get_Vars%:-52%/curve_velocityGc/curve_velo
city%:-14%3curve_accelerationGc3curve_acceleration$:-3%$dotGc$dot%:-19%&evallGc
&evall%:-20%$lenGc$len%:-27%,map_unapplyGc,map_unapply%:-506#/%+modulenameG%)ve
c_calcG
M6R0
I%:-47R6&'%"FG<$%'vectorG%%listG'%"rGF''%*inner_rngG/%%nameG%&rangeG'%*outer_rn
gGF.6(%%compG%*ournormalG%#d1G%#d2G%$ruvG%$spsG6#%[oCopyright~2001~by~Art~Belmo
nte~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$FQF_o9&9'F<F<6*%&crossGc&cross$:-1%$dotGc$dot%:-19%#ssGc#ss%:-43%,
MultipleintGc,Multipleint%:-346#/%+modulenameG%)vec_calcG
M6R0
I%:-37R6#%&thetaG6"6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin
,~Math,~TAMUGF&C$,$*&9$"""%#PiG!"""$!=@%-%(hastypeG6$F,%&floatG-%&evalfG6#%"%GF
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M6R0
I%:-27R6#'%"vG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Phi
lip~B.~Yasskin,~Math,~TAMUGF*-%%sqrtG6$-%$addG6$*$&9$6#%"iG""#/F7;"""-%(vectdim
G6#F5%)symbolicGF*F*F*6#/%+modulenameG%)vec_calcG
M6R0
I%:-17R6#%&thetaG6"6#%^oCopyright~1995,96~by~Art~Belmonte~and~Philip~B.~Yasskin
,~Math,~TAMUGF&C$,$*&9$"""%#PiGF-#F-"$!=@%-%(hastypeG6$F,%&floatG-%&evalfG6#%"%
GF9F&F&F&6#/%+modulenameG%)vec_calcG
M6R0
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$:-6/%#CtGc.curve
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rect2sph%:-40/%$s2cGc(sph2cyl%:-41/%$c2sGc(cyl2sph%:-16/%&MuintGc,Multipleint%:
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ctor%:-466#/%+modulenameG%)vec_calcG
M6R0
I%:-38R6#'%#rcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
ilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""$YQYThe~argument~must~be~a~
list~or~vector~with~3~components.6"7%-T#6#&F36#;"""""#-%)simplifyG6#-%'arctanG6
$&F36#F@&F36#F?&F36#F4F*F*6$%$lenGc$len%:-276#/%+modulenameG%)vec_calcG
M6R0
I%:-28R6%%"fG'%"rG<$%'vectorG%%listG'%$rngG/%%nameG%&rangeG6%%"tG%%compG%&speed
G6#%[oCopyright~1996~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-
%$lhsG6#9&>8%-T#6#-9$6#-%#opG6#-9%6#F9>8&-FA6#-T%6#--%"DG6#FJFK@%/&9"6#!""%%ste
pG-FA6#-T'6%*&F?"""FMF^oF=Fgn-FA6#-F[o6$F]oF=F6F66(%#ssGc#ss%:-43%$lenGc$len%:-
27%,multipleintGc,multipleint%:-336#/%+modulenameG%)vec_calcG
M6R0
I%:-18R6#'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*6&%%varsG
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n,~Math,~TAMUG6"C'@%/9#""#C$>8$&9"6#F=@$4-%%typeG6$F@-F)6#F.YQVThe~second~argum
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$<#""!@'/-%%nopsG6#F@Fhn-%'RETURNG6#Fhn2"""F[oYQguThe~variables~were~different~
for~different~components|+~or~Maple's~automatic~simplification~changed~the~vari
ables.|+~You~may~specify~a~list~of~variables~as~a~second~argument.FM>F@-%#opGF]
o>8%F[o@$0-F\o6#FfnFioYQgnThe~number~of~components~must~equal~the~number~of~var
iables.FM>8&Ffo-%(unapplyG6$-%)simplifyG6#-%$addG6$--&%"DG6#8'6#&FfnF_q6#Fap/F`
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M6R0
I%:-49R6$'%"FG<$-%%listG6#<%%)constantG%*procedureG%%nameG-%'vectorGF*%"fG6$%%v
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~TAMUG6"C%@%/9#""$C$>8$&9"6#F<@$4-%%typeG6$F?-F)6#F.YQUThe~third~argument~must~
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%nopsG6#F?"""YQgpCould~not~determine~the~variables.|+~You~may~specify~a~list~of
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FdoC$>FY-T%6$F?-%(convertG6$-%%evalGFXF)%%trueG%&falseGF7F76&%)Get_VarsGc)Get_V
ars%:-52%-makefunctionGc-makefunction%:-326#/%+modulenameG%)vec_calcG
M6R0
I%:-39R6#'%#rcG<$%'vectorG%%listG6"6#%]oCopyright~1995-7~by~Art~Belmonte~and~Ph
ilip~B.~Yasskin,~Math,~TAMUGF*C$@$0-%(vectdimG6#9$""#YQYThe~argument~must~be~a~
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#"""F*F*6$%$lenGc$len%:-276#/%+modulenameG%)vec_calcG