TD:\Phil\Maple\VecCalc8\lib/VecCalc.............................................................................................................................................................................................................................ûa?VecCalc............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................001M7R0 I(VecCalc`6boc%init$:-1c(VCalias$:-2c,simplifyvec$:-3c&value$:-4c&Limit$:-5c&li mit$:-6c%Diff$:-7c%diff$:-8c$Int$:-9c$int%:-10c"D%:-11c-MakeFunction%:-12c$&->% :-13c-evalFunction%:-14c#&@%:-15c$Dot%:-16c#&.%:-17c'Length%:-18c&Angle%:-19c&C ross%:-20c#&x%:-21c,Determinant%:-22cBLeadingPrincipalMinorDeterminants%:-23c(d eg2rad%:-24c(rad2deg%:-25c+polar2rect%:-26c+rect2polar%:-27c)cyl2rect%:-28c)rec t2cyl%:-29c)sph2rect%:-30c)rect2sph%:-31c(sph2cyl%:-32c(cyl2sph%:-33c.CurveVelo city%:-34c2CurveAcceleration%:-35c*CurveJerk%:-36c-CurveTangent%:-37c,CurveNorm al%:-38c.CurveBinormal%:-39c/CurveCurvature%:-40c-CurveTorsion%:-41c+CurveSpeed %:-42c/CurveArcLength%:-43cG%-evalFunctionG%#&@G%$DotG%#&.G%'Len gthG%&AngleG%&CrossG%#&xG%,DeterminantG%BLeadingPrincipalMinorDeterminantsG%(de g2radG%(rad2degG%+polar2rectG%+rect2polarG%)cyl2rectG%)rect2cylG%)sph2rectG%)re ct2sphG%(sph2cylG%(cyl2sphG%.CurveVelocityG%2CurveAccelerationG%*CurveJerkG%-Cu rveTangentG%,CurveNormalG%.CurveBinormalG%/CurveCurvatureG%-CurveTorsionG%+Curv eSpeedG%/CurveArcLengthG%8$-%(convertG6$9%F/>8%-%$lhsG6#9&> 8&-%)assumingG6$7#-%)simplifyG6#-9$6#-%#opG6#-FB6#FH7#%%realG>8'-T#6$FfnFH@%/&9 "6#!"".%%stepG-T%6%*&FN"""F[oF[pFLFeo-Fho6$FjoFLF?F?6&%+CurveSpeedGc+CurveSpeed %:-42%,multipleintGc,multipleint%:-626$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-54f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-.%'VectorGF*6&%#FlG %%varsG%%nvarG%"iG6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yasski n,~Math,~TAMUG6"C(>8$-%(convertG6$9$F)@'/9#""#@''&9"6#FD<$F,-F)6#F,>8%FG'FG-F0F L>FN-F>6$FGF)YQ]pThe~second~argument~must~be~a~variable~name~or~a~list~or~Vecto r~of~variable~names.F92FDFC@%'7#&FH6#;FDFCFK>FNFYYQjqAfter~the~first~argument,~ the~remaining~arguments~must~be~variable~names~or~a~list~or~Vector~of~variable~ names.F9C$>FN-%&minusG6$<#-%$seqG6$-T#6#%"fG/FfoF<<#""!@%/-%%nopsG6#FN""">FN-%# opGF^pYQerCould~not~determine~the~variables.|+~You~may~specify~the~variables~as ~additional~arguments~or~as~a~list~or~Vector~argument.F9>8&F\p@$0-F]p6#FFNFap-%(unapply G6$-%)simplifyG6#-%$addG6$--&T%6#8'6#&F8$-%(convertG6$9$F(@$ 0-%%nopsG6#F3""$YQ[oThe~argument~must~be~a~list,~vector~or~Vector~with~3~compon ents.F0>8%7%*&&F36#"""FF-%$sinG6#&F36#F=FF&F36#""#*&FDFF-%$cosGFIFF-F56$FA-%)wh attypeG6#-%%evalG6#F7F0F0F06$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-22f*6#'%"AG<$%'MatrixG%)listlistG6#%#AMG6#%`oCopyright~1995-2003~by~Art~Bel monte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%>8$-%(convertG6$9$F(@$4-%%typeG6$F1 -.F(6#%'squareGY6$Q>The~matrix~%1~must~be~square.F.F5-_%.LinearAlgebraG%,Determ inantG6#F1F.F.F.6$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-12f*6$'%%varsG<%%%nameG-%%listG6#F(-%'VectorGF+%&exprsG6#%%exprG6#%`oCopyri ght~1995-2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%>8$-T#6#9%@ $4-%%typeG6$F6<(%*algebraicG-F*6#FA-%)listlistGFC-.F-FC-.%'MatrixGFC-.%&ArrayGF CY6$QjrMakeFunction~expects~its~second~argument~to~be~an~algebraic~expression,~ list,~listlist,~Vector,~Matrix~or~Array~but~received~%1F3F:-T%6$F6-%#opG6#-%(co nvertG6$9$F*F3F36&%,simplifyvecGc,simplifyvec$:-3%,map_unapplyGc,map_unapply%:- 736$%*protectedG/%+modulenameG%(VecCalcG M7R0 I$:-8f*6#%&exprsG6"6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yassk in,~Math,~TAMUGF&@%-%%typeG6$9$<'.%'MatrixG.%%listG.%&arrayG.%'VectorG.%&ArrayG -%$mapG6$9!9"@%-F+6$F<%(indexedG-&T#6#-%#opG6#F<6#F=-FDFIF&F&6$%(oldfuncGc(oldf unc%:-786%%*protectedGFO/%+modulenameG%(VecCalcG M7R0 I%:-75f*6"6$%"iG%'Mu_intG6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B. ~Yasskin,~Math,~TAMUGF$C$>8%&9"6#"""?(8$""#F09#%%trueG>F,-T#6$F,&F.6#F2F$F$6$%$ IntGc$Int$:-96#/%+modulenameG%(VecCalcG M7R0 I%:-65f*6%'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-.%'VectorGF*'%"rGF' '%$rngG/F,%&rangeG6'%#rlG%"tG%#FlG%%compG%%veloG6#%`oCopyright~1996-2003~by~Art ~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C(>8$-%(convertG6$9%F)>8%-%$lhsG6 #9&>8&-FE6$9$F)>8'-%)assumingG6$7#-%)simplifyG6#-FO6#-%#opG6#-FC6#FI7#%%realG>8 (--T#6#FCF\o@%/&9"6#!"".%%stepG-T'6%-T%6$FTF`oFMF[p-F^p6$F`pFMF@F@6(%"DGc"D%:-1 1%$DotGc$Dot%:-16%,MultipleintGc,Multipleint%:-616$%*protectedG/%+modulenameG%( VecCalcG M7R0 I%:-55f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*6&%$argG %(outtypeG%#FlG%%varsG6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Ya sskin,~Math,~TAMUG6"C(@%-%'memberG6$&9"6#9#<&.&F06#%$rowG.F).F0.&F06#%'columnGC $>8$&F?6#;"""!"#>8%F>C$>FOF?>FV-T#6#9$>8&-%(convertG6$FgnF)@'/-%%nopsG6#7#FO""# @''&FO6#Fco<$F,-F)6#F,>8'Ffo'Ffo-FHFjo>F\p-F[o6$FfoF)YQ]pThe~second~argument~mu st~be~a~variable~name~or~a~list~or~Vector~of~variable~names.F82FcoF_o@%'7#&FO6# ;Fco!""Fio>F\pFgpYQjqAfter~the~first~argument,~the~remaining~arguments~must~be~ variable~names~or~a~list~or~Vector~of~variable~names.F8C$>F\p-%&minusG6$<#-%$se qG6$-T%6#%"fG/F[rFin<#""!@%/-F`o6#F\pFS>F\p-%#opGFbrYQerCould~not~determine~the ~variables.|+~You~may~specify~the~variables~as~additional~arguments~or~as~a~lis t~or~Vector~argument.F8@$50-F`o6#Fin""$0FarF]sYQLThere~must~be~3~components~and ~3~variables.F8-%)simplifyG6#-7%,&-&T'Fgo6#&Fgn6#F]sFS-&FisF\t6#&FgnFgoF[q,&-F^ t6#&Fgn6#FSFS-&FisFetFjsF[q,&-FgtF_tFS-FhsFctF[q6#Fdr-T)6$F\p-F[o6$%"%GFVF8F86* %+getvectypeGc+getvectype%:-71%)Get_VarsGc)Get_Vars%:-74%"DGc"D%:-11%-MakeFunct ionGc-MakeFunction%:-126$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-45f*6$%"rG'%"tG%%nameG6#%#rvG6$%)rememberG%`oCopyright~1995-2003~by~Art~Bel monte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%@$4-%'memberG6$-T#6#9$<%%%listG&%'V ectorG6#%$rowG&F<6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or ~Vector.F.F8>8$-%)simplifyGF7-FH6#-T)6$-T%6$FF9%-T'FPF.F.6*%+getvectypeGc+getve ctype%:-71%2CurveAccelerationGc2CurveAcceleration%:-35%,CurveNormalGc,CurveNorm al%:-38%$DotGc$Dot%:-166$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-43f*6$%"rG'%"tG%%nameG6$%"aG%"bG6$%)rememberG%`oCopyright~1995-2003~by~Art~ Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C$@$4-%'memberG6$-T#6#9$<%%%listG& %'VectorG6#%$rowG&F=6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list ~or~Vector.F/F9-%(unapplyG6%-T'6$-T%6$F99%/FO;8$8%FRFSF/F/6(%+getvectypeGc+getv ectype%:-71%+CurveSpeedGc+CurveSpeed%:-42%$IntGc$Int$:-96$%*protectedG/%+module nameG%(VecCalcG M7R0 I%:-33f*6#'%#ccG<%%%listG%'VectorG%'vectorG6$%$cclG%$sclG6#%`oCopyright~1995-20 03~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-%(convertG6$9$F(@$ 0-%%nopsG6#F3""$YQ[oThe~argument~must~be~a~list,~vector~or~Vector~with~3~compon ents.F0>8%7%-T#6#7$&F36#"""&F36#F=&F36#""#-%)simplifyG6#-%'arctanG6$FGFJ-F56$FA -%)whattypeG6#-%%evalG6#F7F0F06$%'LengthGc'Length%:-186$%*protectedG/%+modulena meG%(VecCalcG M7R0 I%:-23f*6#'%"HG<$%'MatrixG%)listlistG6%%#HMG%"nG%"kG6#%`oCopyright~1995-2003~by ~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-%(convertG6$9$F(>8%-%#o pG6$7$"""F>F3@$0F9-F;6$7$F>""#F3YQ;The~matrix~must~be~square.F0-%$seqG6$-_%.Lin earAlgebraG%,DeterminantG6#-_FL%*SubMatrixG6%F3;F>8&FS/FT;F>F9F0F0F06$%*protect edG/%+modulenameG%(VecCalcG M7R0 I%:-13c-MakeFunction%:-126$%*protectedG/%+modulenameG%(VecCalcG M7R0 I$:-9f*6#%&exprsG6"6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yassk in,~Math,~TAMUGF&@%-%%typeG6$9$<'.%'MatrixG.%%listG.%&arrayG.%'VectorG.%&ArrayG -%$mapG6$9!9"@%-F+6$F<%(indexedG-&T#6#-%#opG6#F<6#F=-FDFIF&F&6$%(oldfuncGc(oldf unc%:-796%%*protectedGFO/%+modulenameG%(VecCalcG M7R0 I%:-66f*6%'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-.%'VectorGF*'%"rGF' '%$rngG/F,%&rangeG6'%#rlG%"tG%#FlG%%compG%%veloG6#%`oCopyright~1996-2003~by~Art ~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C(>8$-%(convertG6$9%F)>8%-%$lhsG6 #9&>8&-FE6$9$F)>8'-%)assumingG6$7#-%)simplifyG6#-FO6#-%#opG6#-FC6#FI7#%%realG>8 (--T#6#FCF\o@%/&9"6#!"".%%stepG-T'6%-T%6$FTF`oFMF[p-F^p6$F`pFMF@F@6(%"DGc"D%:-1 1%$DotGc$Dot%:-16%,multipleintGc,multipleint%:-626$%*protectedG/%+modulenameG%( VecCalcG M7R0 I%:-56f*6#'%'fprocsG<)%%listG%%nameG%'MatrixG%'VectorG%&ArrayG%)constantG%*proc edureG6%%%varsG%%nvarG%"iG6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B .~Yasskin,~Math,~TAMUG6"@%-%%typeG6$9$<&F(F*F+F,-%$mapG6%T#F:&9"6#;""#9#C&@'/FE FD@''&FA6#FD<$F)-F(6#F)>8$FK'FK-.F+FO>FQ-%(convertG6$FKF(YQ]pThe~second~argumen t~must~be~a~variable~name~or~a~list~or~Vector~of~variable~names.F52FDFE@%'7#F@F N>FQFhnYQjqAfter~the~first~argument,~the~remaining~arguments~must~be~variable~n ames~or~a~list~or~Vector~of~variable~names.F5C$>FQ-%&minusG6$<#-T%6#F:<#""!@%/- %%nopsG6#FQ""">FQ-%#opGF[pYQerCould~not~determine~the~variables.|+~You~may~spec ify~the~variables~as~additional~arguments~or~as~a~list~or~Vector~argument.F5>8% Fio>FQF^p-%(unapplyG6$-%)simplifyG6#-%$addG6$--&T'6$8&FcqFdoF[p/Fcq;F\pFcpFQF5F 56(%*LaplacianGc*Laplacian%:-56%)Get_VarsGc)Get_Vars%:-74%"DGc"D%:-116$%*protec tedG/%+modulenameG%(VecCalcG M7R0 I%:-46f*6"F$6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasskin,~Mat h,~TAMUGF$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$T7F,-F)6$T9F,F$F$6:%.CurveVelocityGc.CurveVe locity%:-34%2CurveAccelerationGc2CurveAcceleration%:-35%*CurveJerkGc*CurveJerk% :-36%-CurveTangentGc-CurveTangent%:-37%,CurveNormalGc,CurveNormal%:-38%.CurveBi normalGc.CurveBinormal%:-39%/CurveCurvatureGc/CurveCurvature%:-40%-CurveTorsion Gc-CurveTorsion%:-41%+CurveSpeedGc+CurveSpeed%:-42%/CurveArcLengthGc/CurveArcLe ngth%:-43%8$-T%F7-%)simplifyG6#-T+6$-T'6$FF9%-T)FQF.F.6,%+getvectypeGc+getve ctype%:-71%,simplifyvecGc,simplifyvec$:-3%2CurveAccelerationGc2CurveAcceleratio n%:-35%-CurveTangentGc-CurveTangent%:-37%$DotGc$Dot%:-166$%*protectedG/%+module nameG%(VecCalcG M7R0 I%:-34f*6$%"rG'%"tG%%nameG6"6$%)rememberG%`oCopyright~1995-2003~by~Art~Belmonte ~and~Philip~B.~Yasskin,~Math,~TAMUGF)C$@$4-%'memberG6$-T#6#9$<%%%listG&%'Vector G6#%$rowG&F:6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or~Vect or.F)F6-%$mapG6%T%-T'F59%F)F)6(%+getvectypeGc+getvectype%:-71%%diffGc%diff$:-8% ,simplifyvecGc,simplifyvec$:-36$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-24f*6#%&thetaG6"6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yass kin,~Math,~TAMUGF&C$,$*&9$"""%#PiGF-#F-"$!=@%-%(hastypeG6$F,%&floatG-%&evalfG6# %"%GF9F&F&F&6$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-14f*6#%"fG6"6#%PCopyright~2003~by~Philip~B.~Yasskin,~Math,~TAMUGF&@)-%%type G6$9$<(%%listG%'MatrixG%&ArrayG&%'VectorG6#%$rowG%)listlistG&F36#%'columnG-%$ma pG6$T#9"2""#9#-F-6#&F>6#;F@FA-%'memberG6$-T%6#&F>6#F@<%F/F2F7-F-6#-%#opG6#-%(co nvertG6$-T'FLF/-F-FLF&F&6(%-evalFunctionGc-evalFunction%:-14%+getvectypeGc+getv ectype%:-71%,simplifyvecGc,simplifyvec$:-36$%*protectedG/%+modulenameG%(VecCalc G M7R0 I%:-76%"DG6" M7R0 I%:-67f*6&'%"fG<%%%nameG%)constantG%*procedureG'%"RG<$-%%listG6#F'-.%'VectorGF0 '%'innrngG/F(%&rangeG'%'outrngGF66)%#RlG%%varsG%$RuvG%%compG%#e1G%#e2G%*lennorm alG6#%`oCopyright~1996-2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6 "C,>8$-%(convertG6$9%F/>8%-%&minusG6$<#-%$seqG6$-T#6#%"cG/FXFG<#""!@%/-%%nopsG6 #FM""">FM-%#opG6#-F^oFjnYQisCould~not~determine~the~variables~for~the~surface.| +~Be~sure~to~specify~the~surface~as~a~list~or~Vector~of~functions~of~the~same~t wo~variables.FD@$0<#FM<$-%$lhsG6#9'-Fho6#9&YQepThe~variables~of~integration~do~ not~match~the~parameters~for~the~surface.~Please~fix~them.FD>8&-FGFjn>8'-%)assu mingG6$7#-%)simplifyG6#-9$6#-F^o6#Fap7#%%realG>8(-T%6$Fap&FM6#F[o>8)-Ffq6$Fap&F M6#""#>8*-T)6#-T'6$FdqF[r@%/&9"6#!"".%%stepG-T+6&*&FdpF[oFbrF[oF]pFjoF_s-Fbs6%F dsF]pFjoFDFD6,%)Get_VarsGc)Get_Vars%:-74%%diffGc%diff$:-8%&CrossGc&Cross%:-20%' LengthGc'Length%:-18%,MultipleintGc,Multipleint%:-616$%*protectedG/%+modulename G%(VecCalcG M7R0 I%:-57f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*6(%$argG %(outtypeG%#FlG%%varsG%"iG%"jG6#%`oCopyright~1996-2003~by~Art~Belmonte~and~Phil ip~B.~Yasskin,~Math,~TAMUG6"C'@)-%'memberG6$&9"6#9#<$.%'MatrixG.%)listlistGC$>8 $&FA6#;"""!"#>8%F@-%%typeG6$-%%evalG6#9$F)C$>FKFA>FRFH-FT6$FVF0C$>FKFA>FRFFC$>F KFA>FR%1OutputMatrixTypeG>8&-%(convertG6$FYF)@'/-%%nopsG6#7#FK""#@''&FK6#F[p<$F ,-F)6#F,>8'F^p'F^p-.F0Fbp>Fdp-Fco6$F^pF)YQ]pThe~second~argument~must~be~a~varia ble~name~or~a~list~or~Vector~of~variable~names.F:2F[pFgo@%'7#&FK6#;F[p!""Fap>Fd pF`qYQjqAfter~the~first~argument,~the~remaining~arguments~must~be~variable~name s~or~a~list~or~Vector~of~variable~names.F:C$>Fdp-%&minusG6$<#-%$seqG6$-T#6#%"fG /FdrFao<#""!@%/-Fho6#FdpFO>Fdp-%#opGF[sYQerCould~not~determine~the~variables.|+ ~You~may~specify~the~variables~as~additional~arguments~or~as~a~list~or~Vector~a rgument.F:-%)simplifyG6#7#-F_r6$7#-F_r6$--&T%6#8(6#&Fao6#8)6#F]s/F_t;FOFjr/Fct; FO-Fho6#Fao-T'6$Fdp-Fco6$%"%GFRF:F:6(%)Get_VarsGc)Get_Vars%:-74%"DGc"D%:-11%-Ma keFunctionGc-MakeFunction%:-126$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-47f*6%%"RG'%"uG%%nameG'%"vGF(6#%#RvG6$%)rememberG%`oCopyright~1995-2003~by~ Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%@$4-%'memberG6$-T#6#9$<%%%li stG&%'VectorG6#%$rowG&F>6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~ list~or~Vector.F0F:>8$-T%F96$-%$mapG6%T'FH9%-FM6%FOFH9&F0F06(%+getvectypeGc+get vectype%:-71%,simplifyvecGc,simplifyvec$:-3%%diffGc%diff$:-86$%*protectedG/%+mo dulenameG%(VecCalcG M7R0 I%:-35f*6$%"rG'%"tG%%nameG6"6$%)rememberG%`oCopyright~1995-2003~by~Art~Belmonte ~and~Philip~B.~Yasskin,~Math,~TAMUGF)C$@$4-%'memberG6$-T#6#9$<%%%listG&%'Vector G6#%$rowG&F:6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or~Vect or.F)F6-%$mapG6%T%-T'6$F69%FJF)F)6(%+getvectypeGc+getvectype%:-71%%diffGc%diff$ :-8%.CurveVelocityGc.CurveVelocity%:-346$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-25f*6#%&thetaG6"6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yass kin,~Math,~TAMUGF&C$,$*&9$"""%#PiG!"""$!=@%-%(hastypeG6$F,%&floatG-%&evalfG6#%" %GF9F&F&F&6$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-15c-evalFunction%:-146$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-77%%DiffG6" M7R0 I%:-68f*6&'%"fG<%%%nameG%)constantG%*procedureG'%"RG<$-%%listG6#F'-.%'VectorGF0 '%'innrngG/F(%&rangeG'%'outrngGF66)%#RlG%%varsG%$RuvG%%compG%#e1G%#e2G%*lennorm alG6#%`oCopyright~1996-2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6 "C,>8$-%(convertG6$9%F/>8%-%&minusG6$<#-%$seqG6$-T#6#%"cG/FXFG<#""!@%/-%%nopsG6 #FM""">FM-%#opG6#-F^oFjnYQisCould~not~determine~the~variables~for~the~surface.| +~Be~sure~to~specify~the~surface~as~a~list~or~Vector~of~functions~of~the~same~t wo~variables.FD@$0<#FM<$-%$lhsG6#9'-Fho6#9&YQepThe~variables~of~integration~do~ not~match~the~parameters~for~the~surface.~Please~fix~them.FD>8&-FGFjn>8'-%)assu mingG6$7#-%)simplifyG6#-9$6#-F^o6#Fap7#%%realG>8(-T%6$Fap&FM6#F[o>8)-Ffq6$Fap&F M6#""#>8*-T)6#-T'6$FdqF[r@%/&9"6#!"".%%stepG-T+6&*&FdpF[oFbrF[oF]pFjoF_s-Fbs6%F dsF]pFjoFDFD6,%)Get_VarsGc)Get_Vars%:-74%%diffGc%diff$:-8%&CrossGc&Cross%:-20%' LengthGc'Length%:-18%,multipleintGc,multipleint%:-626$%*protectedG/%+modulename G%(VecCalcG M7R0 I%:-58f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*6(%#FlG% %varsG%&ncompG%%nvarG%"iG%"jG6#%`oCopyright~1996-2003~by~Art~Belmonte~and~Phili p~B.~Yasskin,~Math,~TAMUG6"C*>8$-%(convertG6$9$F)@'/9#""#@''&9"6#FE<$F,-F)6#F,> 8%FH'FH-.F0FM>FO-F?6$FHF)YQ]pThe~second~argument~must~be~a~variable~name~or~a~l ist~or~Vector~of~variable~names.F:2FEFD@%'7#&FI6#;FEFDFL>FOFenYQjqAfter~the~fir st~argument,~the~remaining~arguments~must~be~variable~names~or~a~list~or~Vector ~of~variable~names.F:C$>FO-%&minusG6$<#-%$seqG6$-T#6#%"fG/FhoF=<#""!@%/-%%nopsG 6#FO""">FO-%#opGF`pYQerCould~not~determine~the~variables.|+~You~may~specify~the ~variables~as~additional~arguments~or~as~a~list~or~Vector~argument.F:>8&-F_p6#F =>8'F^p@$0FhpF\qYQgnThe~number~of~components~must~equal~the~number~of~variables .F:>FOFcp-F?6$7#-Fco6$7#-Fco6$--&T%6#8(6#&F=6#8)F`p/F_r;FapF\q/Fcr;FapFhp%'Matr ixG-%(unapplyG6$-%)simplifyG6#-_%.LinearAlgebraG%,DeterminantG6#%"%GFOF:F:6&%)G et_VarsGc)Get_Vars%:-74%"DGc"D%:-116$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-48f*6%%"RG'%"uG%%nameG'%"vGF(6"6$%)rememberG%`oCopyright~1995-2003~by~Art~B elmonte~and~Philip~B.~Yasskin,~Math,~TAMUGF+-T%6#-T#6%9$9%9&F+F+6&%0SurfaceTang entsGc0SurfaceTangents%:-47%&CrossGc&Cross%:-206$%*protectedG/%+modulenameG%(Ve cCalcG M7R0 I%:-36f*6$%"rG'%"tG%%nameG6"6$%)rememberG%`oCopyright~1995-2003~by~Art~Belmonte ~and~Philip~B.~Yasskin,~Math,~TAMUGF)C$@$4-%'memberG6$-T#6#9$<%%%listG&%'Vector G6#%$rowG&F:6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or~Vect or.F)F6-%$mapG6%T%-T'6$F69%FJF)F)6(%+getvectypeGc+getvectype%:-71%%diffGc%diff$ :-8%2CurveAccelerationGc2CurveAcceleration%:-356$%*protectedG/%+modulenameG%(Ve cCalcG M7R0 I%:-26f*6#'%#pcG<%%%listG%'VectorG%'vectorG6$%$pclG%$rclG6#%`oCopyright~1995-20 03~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-%(convertG6$9$F(@$ 0-%%nopsG6#F3""#YQ[oThe~argument~must~be~a~list,~vector~or~Vector~with~2~compon ents.F0>8%7$*&&F36#"""FF-%$cosG6#&F36#F=FF*&FDFF-%$sinGFIFF-F56$FA-%)whattypeG6 #-%%evalG6#F7F0F0F06$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-16f*6$%"vG%"wG6&%#vlG%#wlG%"nG%"iG6#%`oCopyright~1995-2003~by~Art~Belmonte~ and~Philip~B.~Yasskin,~Math,~TAMUG6"C(@&4-%'memberG6$-T#6#9$<%%%listG&%'VectorG 6#%$rowG&F<6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or~Vecto r.F.F84-F36$-F66#9%F9Y6$FDFJ>8$-%(convertG6$-T%F7F:>8%-FP6$-FSFIF:>8&-%%nopsG6# FN@$0FZ-Ffn6#FUYQYThe~two~vectors~must~have~the~same~number~of~components.F.-%) simplifyG6#-%$addG6$*&&FN6#8'"""&FUFfoFho/Fgo;FhoFZF.F.6&%+getvectypeGc+getvect ype%:-71%,simplifyvecGc,simplifyvec$:-36$%*protectedG/%+modulenameG%(VecCalcG M7R0 I$:-1f*6"F$F$F$C,>%1OutputVectorTypeG.&%'VectorG6#%$rowG>%1OutputMatrixTypeG.%' MatrixG-%'printfG6$%$%s|+G%NInitializing~Package:~~~VecCalc~~~Version~8.0G-F26$ F4%AAll~HELP~is~linked~from~?VecCalcG-F26$F4%DTo~use~aliases,~execute:~~~VCalia s;G-F26$F4%DLoading~packages:~~~student~~~plotsG-%*interfaceG6#/%*warnlevelG""! -%%withG6#%(studentG-FF6#%&plotsG-F@6#/FC""$F$6$F'F.F$6$%*protectedG/%+modulena meG%(VecCalcG M7R0 I%:-78%%diffG6" M7R0 I%:-69f*6&'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*'%"RG<$F (-.F0F*'%'innrngG/F,%&rangeG'%'outrngGF86*%#RlG%%varsG%$RuvG%#FlG%%compG%#e1G%# e2G%*normalvecG6#%`oCopyright~1996-2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~ Math,~TAMUG6"C->8$-%(convertG6$9%F)>8%-%&minusG6$<#-%$seqG6$-T#6#%"cG/FenFJ<#"" !@%/-%%nopsG6#FP""">FP-%#opG6#-FaoF]oYQisCould~not~determine~the~variables~for~ the~surface.|+~Be~sure~to~specify~the~surface~as~a~list~or~Vector~of~functions~ of~the~same~two~variables.FG@$0<#FP<$-%$lhsG6#9'-F[p6#9&YQepThe~variables~of~in tegration~do~not~match~the~parameters~for~the~surface.~Please~fix~them.FG>8&-FJ F]o>8'-FL6$9$F)>8(-%)assumingG6$7#-%)simplifyG6#-Fgp6#-Fao6#Fdp7#%%realG>8)-T%6 $Fdp&FP6#F^o>8*-F]r6$Fdp&FP6#""#>8+-T'6$F[rFbr@%/&9"6#!"".%%stepG-T+6&-T)6$F\qF irF`pF]pFcs-Ffs6%FhsF`pF]pFGFG6,%)Get_VarsGc)Get_Vars%:-74%%diffGc%diff$:-8%&Cr ossGc&Cross%:-20%$DotGc$Dot%:-16%,MultipleintGc,Multipleint%:-616$%*protectedG/ %+modulenameG%(VecCalcG M7R0 I%:-59f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*6)%#FlG% %varsG%"nG%"iG%"jG%"fG%#dgG6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~ B.~Yasskin,~Math,~TAMUG6"C,>8$-%(convertG6$9$F)@'/9#""#@''&9"6#FF<$F,-F)6#F,>8% FI'FI-.F0FN>FP-F@6$FIF)YQ]pThe~second~argument~must~be~a~variable~name~or~a~lis t~or~Vector~of~variable~names.F;2FFFE@%'7#&FJ6#;FFFEFM>FPFfnYQjqAfter~the~first ~argument,~the~remaining~arguments~must~be~variable~names~or~a~list~or~Vector~o f~variable~names.F;C$>FP-%&minusG6$<#-%$seqG6$-T#6#8)/FioF><#""!@%/-%%nopsG6#FP """>FP-%#opGFapYQerCould~not~determine~the~variables.|+~You~may~specify~the~var iables~as~additional~arguments~or~as~a~list~or~Vector~argument.F;>8&F_p@$0-F`p6 #F>FipYQgnThe~number~of~components~must~equal~the~number~of~variables.F;>F>-F>6 #Fdp?(8'FbpFbp,&FipFbp!""Fbp%%trueG?(8(,&FdqFbpFbpFbpFbpFipFgq@$0-%)simplifyG6# ,&-T%6$&F>6#Fdq&FP6#FiqFbp-Fbr6$&F>Fgr&FPFerFfqF\pO.%*undefinedG>Fio-T'6$&F>6#F bp&FPFds?(FdqFFFbpFipFgqC$>8*-F^r6#,&FdrFbp-Fbr6$FioF[sFfq@%-%$hasG6$Fis&FP6#;F bp,&FdqFbpFfqFbpOF]s>Fio,&FioFbp-Fas6$FisF[sFbp?(FdqFbpFbpFipFgq@$0FjsF\pYQGUna ble~to~compute~necessary~integrals.F;-T)6$FPFioF;F;6*%)Get_VarsGc)Get_Vars%:-74 %%diffGc%diff$:-8%$intGc$int%:-10%-MakeFunctionGc-MakeFunction%:-126$%*protecte dG/%+modulenameG%(VecCalcG M7R0 I%:-49f*6%%"RG'%"uG%%nameG'%"vGF(6"6$%)rememberG%`oCopyright~1995-2003~by~Art~B elmonte~and~Philip~B.~Yasskin,~Math,~TAMUGF+-T%6#-T#6%9$9%9&F+F+6&%.SurfaceNorm alGc.SurfaceNormal%:-48%'LengthGc'Length%:-186$%*protectedG/%+modulenameG%(VecC alcG M7R0 I%:-37f*6$%"rG'%"tG%%nameG6#%"vG6$%)rememberG%`oCopyright~1995-2003~by~Art~Belm onte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C%@$4-%'memberG6$-T#6#9$<%%%listG&%'Ve ctorG6#%$rowG&F<6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~list~or~ Vector.F.F8>8$-T%6$F89%-%)assumingG6$7#-T)6#*&FF"""-T'6#FF!""7#%%realGF.F.6*%+g etvectypeGc+getvectype%:-71%.CurveVelocityGc.CurveVelocity%:-34%'LengthGc'Lengt h%:-18%,simplifyvecGc,simplifyvec$:-36$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-27f*6#'%#rcG<%%%listG%'VectorG%'vectorG6$%$rclG%$pclG6#%`oCopyright~1995-20 03~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-%(convertG6$9$F(@$ 0-%%nopsG6#F3""#YQ[oThe~argument~must~be~a~list,~vector~or~Vector~with~2~compon ents.F0>8%7$-T#F<-%)simplifyG6#-%'arctanG6$&F36#F=&F36#"""-F56$FA-%)whattypeG6# -%%evalG6#F7F0F06$%'LengthGc'Length%:-186$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-17c$Dot%:-166$%*protectedG/%+modulenameG%(VecCalcG M7R0 I$:-2-%&aliasG6T/%#MFGc-MakeFunction%:-12/%#EFGc-evalFunction%:-14/%$DetGc,Dete rminant%:-22/%%LPMDGcBLeadingPrincipalMinorDeterminants%:-23/%$d2rGc(deg2rad%:- 24/%$r2dGc(rad2deg%:-25/%$p2rGc+polar2rect%:-26/%$r2pGc+rect2polar%:-27/%$c2rGc )cyl2rect%:-28/%$r2cGc)rect2cyl%:-29/%$s2rGc)sph2rect%:-30/%$r2sGc)rect2sph%:-3 1/%$s2cGc(sph2cyl%:-32/%$c2sGc(cyl2sph%:-33/%#CvGc.CurveVelocity%:-34/%#CaGc2Cu rveAcceleration%:-35/%#CjGc*CurveJerk%:-36/%#CsGc+CurveSpeed%:-42/%#CLGc/CurveA rcLength%:-43/%#CTGc-CurveTangent%:-37/%#CNGc,CurveNormal%:-38/%#CBGc.CurveBino rmal%:-39/%#CkGc/CurveCurvature%:-40/%#CtGc-CurveTorsion%:-41/%$CaTGc8$-%(convertG6$9%F)>8%-%&minusG6$<#-%$seqG6$-T#6#%"cG/FenFJ<#"" !@%/-%%nopsG6#FP""">FP-%#opG6#-FaoF]oYQisCould~not~determine~the~variables~for~ the~surface.|+~Be~sure~to~specify~the~surface~as~a~list~or~Vector~of~functions~ of~the~same~two~variables.FG@$0<#FP<$-%$lhsG6#9'-F[p6#9&YQepThe~variables~of~in tegration~do~not~match~the~parameters~for~the~surface.~Please~fix~them.FG>8&-FJ F]o>8'-FL6$9$F)>8(-%)assumingG6$7#-%)simplifyG6#-Fgp6#-Fao6#Fdp7#%%realG>8)-T%6 $Fdp&FP6#F^o>8*-F]r6$Fdp&FP6#""#>8+-T'6$F[rFbr@%/&9"6#!"".%%stepG-T+6&-T)6$F\qF irF`pF]pFcs-Ffs6%FhsF`pF]pFGFG6,%)Get_VarsGc)Get_Vars%:-74%%diffGc%diff$:-8%&Cr ossGc&Cross%:-20%$DotGc$Dot%:-16%,multipleintGc,multipleint%:-626$%*protectedG/ %+modulenameG%(VecCalcG M7R0 I%:-60f*6#'%"FG<$-%%listG6#<%%%nameG%)constantG%*procedureG-%'VectorGF*6*%$argG %(outtypeG%#FlG%%varsG%"nG%"iG%"jG%"AG6#%`oCopyright~1995-2003~by~Art~Belmonte~ and~Philip~B.~Yasskin,~Math,~TAMUG6"C-@%-%'memberG6$&9"6#9#<&.&F06#%$rowG.F).F0 .&F06#%'columnGC$>8$&FC6#;"""!"#>8%FBC$>FSFC>FZ-%)whattypeG6#-%%evalG6#9$>8&-%( convertG6$F^oF)@'/-%%nopsG6#7#FS""#@''&FS6#Fjo<$F,-F)6#F,>8'F]p'F]p-FLFap>Fcp-F bo6$F]pF)YQ]pThe~second~argument~must~be~a~variable~name~or~a~list~or~Vector~of ~variable~names.F<2FjoFfo@%'7#&FS6#;Fjo!""F`p>FcpF^qYQjqAfter~the~first~argumen t,~the~remaining~arguments~must~be~variable~names~or~a~list~or~Vector~of~variab le~names.FFcp-%&minusG6$<#-%$seqG6$-T#6#%"fG/FbrF`o<#""!@%/-Fgo6#FcpFW>Fcp- %#opGFirYQerCould~not~determine~the~variables.|+~You~may~specify~the~variables~ as~additional~arguments~or~as~a~list~or~Vector~argument.F<@$50-Fgo6#F`o""$0FhrF dsYQLThere~must~be~3~components~and~3~variables.F<>F`o-F`o6#F[s@$0-%)simplifyG6 #-%$addG6$-T%6$&F`o6#8)&FcpFgt/Fht;FWFdsFerO.%*undefinedG>8+7%FerFerFer>&F`u6#F W,&-T'6$&F`oF^p&Fcp6#FdsFW-Fgu6$,&-Fgu6$-Fdt6$&F`oF[vFjuFjuFWFcvFbq&FcpF^pFW>&F `uF^p,$-Fgu6$&F`oFduFjuFbq@$550-F^t6#,(-Fdt6$&F`uF[vFdvFW-Fdt6$FfvFjuFbqFjvFbqF er0-F^t6#,(-Fdt6$FcuFjuFW-Fdt6$Fdw&FcpFduFbqFiuFbqFer0-F^t6#,(-Fdt6$FfvF_xFW-Fd t6$FcuFdvFbqFcvFbqFerYQGUnable~to~compute~necessary~integrals.F<-T)6$Fcp-Fbo6$F `uFZF8$-T%F<>8%-%(convertG6$FKFA>8&-T'6$FO9%>8'-T )FW>8(-%)simplifyG6#,&FZ"""*(-T+6$FZFTF]o-F`o6$FTFT!""FTF]oFdo>8)-%)assumingG6$ 7#-Fjn6#*&FhnF]o-T-6#FhnFdo7#%%realG-FQ6$Ffo-%)whattypeG6#-%%evalG6#FKF3F36.%+g etvectypeGc+getvectype%:-71%,simplifyvecGc,simplifyvec$:-3%.CurveVelocityGc.Cur veVelocity%:-34%2CurveAccelerationGc2CurveAcceleration%:-35%$DotGc$Dot%:-16%'Le ngthGc'Length%:-186$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-28f*6#'%#ccG<%%%listG%'VectorG%'vectorG6$%$cclG%$rclG6#%`oCopyright~1995-20 03~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C&>8$-%(convertG6$9$F(@$ 0-%%nopsG6#F3""$YQ[oThe~argument~must~be~a~list,~vector~or~Vector~with~3~compon ents.F0>8%7%*&&F36#"""FF-%$cosG6#&F36#""#FF*&FDFF-%$sinGFIFF&F36#F=-F56$FA-%)wh attypeG6#-%%evalG6#F7F0F0F06$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-18f*6#%"vG6"6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yasskin, ~Math,~TAMUGF&-%)assumingG6$7#-%)simplifyG6#-%%sqrtG6#-T#6$9$F67#%%realGF&F&6$% $DotGc$Dot%:-166$%*protectedG/%+modulenameG%(VecCalcG M7R0 I$:-3f*6#%"vG6#%&vtypeG6#%PCopyright~2003~by~Philip~B.~Yasskin,~Math,~TAMUG6"C$ >8$-T#6#9$@'-%'memberG6$F-<$%%listG%)listlistG-%(convertG6$-%&evalmGF0F--F46$F- <&%'MatrixG%&ArrayG&%'VectorG6#%$rowG&FD6#%'columnG-%)simplifyGF0F1F*F*6$%+getv ectypeGc+getvectype%:-716$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-71f*6#%%exprG6&%"QG%%itemG%)vectypesG%"tG6#%BCopyright~2003~by~Jeffrey~Yass kinG6"C&>8&7(%)listlistG%%listG&%'VectorG6#%$rowG&F56#%'columnG%'MatrixG%&Array G>8$-%,SimpleQueueG6#9$?(F-"""FDF-4-_F>%&emptyGF-C&>8%-_F>%(dequeueGF-?&8'F0%%t rueG@$-%%typeG6$FKFPOFP@$-FT6$FK%&arrayGYQfnThis~command~cannot~operate~on~vect ors,~matrices~or~arrays.F-@$4-FT6$FK%'atomicG-%$mapG6$f*6#%"xGF-6$%)operatorG%& arrowGF--_T#%(enqueueGFAF-F-6$F'F>7#-%#opG6#-%%evalG6#FKO%&falseGF-F-F-6#/%+mod ulenameG%(VecCalcG M7R0 I%:-61f*6"F$6#%`oCopyright~1995-2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~Mat h,~TAMUGF$C'@$29#""#YQXAn~integral~needs~at~least~one~variable~of~integration.F $@$0&9"6#!""%%stepG@%-%%typeG6$7#&F16#;F+F3-%%listG6#<$%%nameG/FA%&rangeGO-T#6# F1YQinEach~variable~of~integration~must~be~a~name~or~a~name~=~range.F$@$4-F76$7 #&F16#;F+!"#-F>6#FBYQUEach~variable~of~integration~must~be~a~name~=~range.F$-T% FG-FF6#&F16#;"""FRF$F$6&%,MuInt_noChkGc,MuInt_noChk%:-75%,multipleintGc,multipl eint%:-626$%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-51f*6"F$6#%`oCopyright~1995-2001~by~Art~Belmonte~and~Philip~B.~Yasskin,~Mat h,~TAMUGF$C&-%'forgetG6$T#9"-F)6$T%F,-F)6$T'F,-F)6$T)F,F$F$6*%0SurfaceTangentsG c0SurfaceTangents%:-47%.SurfaceNormalGc.SurfaceNormal%:-48%4SurfaceNormalLength Gc4SurfaceNormalLength%:-49%,SurfaceAreaGc,SurfaceArea%:-506$%*protectedG/%+mod ulenameG%(VecCalcG M7R0 I%:-39f*6$%"rG'%"tG%%nameG6&%#rvG%#rVG%#cpG%#bVG6$%)rememberG%`oCopyright~1995- 2003~by~Art~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C)@$4-%'memberG6$-T#6# 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bles~as~additional~arguments~or~as~a~list~or~Vector~argument.F4>8'F_q-%)simplif yG6#7#-%$seqG6$7#-F]r6$--&T%6$8(8)Fip6#Fbq/Fhr;FIFgq/FgrF[s-T'6$F[o-Fbo6$%"%GFL F4F46(%)Get_VarsGc)Get_Vars%:-74%"DGc"D%:-11%-MakeFunctionGc-MakeFunction%:-126 $%*protectedG/%+modulenameG%(VecCalcG M7R0 I%:-41f*6$%"rG'%"tG%%nameG6$%#rVG%#cpG6$%)rememberG%`oCopyright~1995-2003~by~Ar t~Belmonte~and~Philip~B.~Yasskin,~Math,~TAMUG6"C'@$4-%'memberG6$-T#6#9$<%%%list G&%'VectorG6#%$rowG&F=6#%'columnGY6$QWThe~argument~%1~does~not~simplify~to~a~li st~or~Vector.F/F9>8$-%(convertG6$-T%F8F=@$0-%#opG6$"""FG""$YQMThe~torsion~is~on ly~defined~in~3~dimensions.F/>8%-T+6$-T'6$FG9%-T)Fgn-%)simplifyG6#*&-T/6$FW-T-F gnFR-F`o6$FWFW!""F/F/60%+getvectypeGc+getvectype%:-71%,simplifyvecGc,simplifyve c$:-3%.CurveVelocityGc.CurveVelocity%:-34%2CurveAccelerationGc2CurveAcceleratio n%:-35%&CrossGc&Cross%:-20%*CurveJerkGc*CurveJerk%:-36%$DotGc$Dot%:-166$%*prote ctedG/%+modulenameG%(VecCalcG M7R0 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