{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 3" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 31 "Line Integrals of Scalar Fields " }}{PARA 18 "" 0 "" {TEXT -1 20 "Using Maple and the " }{TEXT 256 8 " vec_calc" }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT -1 74 "This w orksheet shows how to compute line integrals of scalar fields using " }{TEXT 257 5 "Maple" }{TEXT -1 9 " and the " }{TEXT 258 8 "vec_calc" } {TEXT -1 33 " package. As examples we compute" }}{PARA 0 "" 0 "" {TEXT 260 38 "* The Average Temperature along a Wire" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 261 39 "* The Mass and Center of Mass of a Wir e" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "To start the " }{TEXT 259 8 "vec_calc" }{TEXT -1 41 " pac kage, execute the following commands:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "libname:=libname,\"C:/mylib/vector\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q=C:\\Program~Files\\Maple~6/lib6\"Q0C:/mylib/vecto rF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_a liases:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n orm and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 " " {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 6 "" 1 "" {TEXT -1 33 "Package: vec_calc Version 4.3" }} {PARA 6 "" 1 "" {TEXT -1 32 "For all HELP, execute: ?vec_calc" }} {PARA 6 "" 1 "" {TEXT -1 38 "To use aliases, execute: vc_aliases;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 " " {TEXT 262 36 "The Average Temperature along a Wire" }{TEXT -1 0 "" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "When a temperature probe is inse rted into a fluid, it records the average temperature along the probe. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The t emperature of the fluid in a 27\"x36\"x21\" box is " }{XPPEDIT 18 0 "T (x,y,z) = x*y-x*z;" "6#/-%\"TG6%%\"xG%\"yG%\"zG,&*&F'\"\"\"F(F,F,*&F'F ,F)F,!\"\"" }{TEXT -1 41 ". A probe having the shape of the curve " } {XPPEDIT 18 0 "r(t) = [3*t^2, t^3+3*t, t^3-3*t+3];" "6#/-%\"rG6#%\"tG7 %*&\"\"$\"\"\"*$F'\"\"#F+,&*$F'F*F+*&F*F+F'F+F+,(*$F'F*F+*&F*F+F'F+!\" \"F*F+" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 < t;" "6#2\"\"!%\"tG" } {TEXT -1 152 " < 3 is inserted into the box. We want to find the aver age temperature along the probe. Then we will compare it to the avera ge tamperature in the box." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We first define the functions which describe the \+ curve and the temperature." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r:=MF(t,[3*t^2, t^3+3*t, t^3-3*t+3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%R6#%\"tG6\"6$%)operatorG%&arrowGF),$*$)9$\"\"# \"\"\"\"\"$F)F)F)RF'F)F*F),&*$)F0F3F2F2*&F3F2F0F2F2F)F)F)RF'F)F*F),(F6 F2*&F3F2F0F2!\"\"F3F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "T:=MF([x,y,z],x*y-x*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TGR 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"9$F1F1*&F2F19&F 1!\"\"F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We can plot the c urve using the command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 " spacecurve(r(t),t=0..3, axes=boxed, color=blue, thickness=3, orientati on=[-30,80], scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%'CURVESG6$7T7%$\"\"!F)F($\"\"$F)7%$\"+X9`C6!# 6$\"+c'H!R=!#5$\"+Fgb;G!\"*7%$\"+#yD\")\\%F/$\"+&3H=p$F2$\"+J!*[MEF57% $\"+,$y?,\"F2$\"+g!o@d&F2$\"+!*f.6#F57%$\"+.KJ[SF2$\"+)*>h^ 6F5$\"+m.`Z>F57%$\"+#3/-^&F2$\"+([JWO\"F5$\"+:H+$z\"F57%$\"+]7+(>(F2$ \"+;)*)oe\"F5$\"+1V6[;F57%$\"+3Zq3\"*F2$\"+kRO?=F5$\"+::C9:F57%$F.F5$ \"+24Bm?F5$\"+>:w#R\"F57%$\"+\\Iog8F5$\"+Bw'eK#F5$\"+(H^]G\"F57%$\"+\" GD$>;F5$\"+*3^1g#F5$\"+Cy[#>\"F57%$\"+V\"e/!>F5$\"+$Ge>*GF5$\"+z![k6\" F57%$\"+L;3/AF5$\"+\"=m6?$F5$\"+Q!4$e5F57%$\"+_d>IDF5$\"+h5F57%$\"++0!)yGF5$\"++?zyQF5$\"+!)4C,5F57%$\"+!)e*)\\KF5$\"+zQ'*\\UF 5$\"+F57%$\"+#fK=g(F5$\"+uc84))F5$\"+__6 eAF57%$\"+hU$y>)F5$\"+owNw%*F5$\"+1.*zb#F57%$\"+flK;))F5$\"+0e2=5!\")$ \"+[P/&*GF57%$\"+&[4tX*F5$\"+!QrB4\"Fjw$\"+`DlqKF57%$\"+/$y?,\"Fjw$\"+ )=g1<\"Fjw$\"++P>'o$F57%$\"+BZn!3\"Fjw$\"+H*zID\"Fjw$\"+mT/VTF57%$\"+/ -_^6Fjw$\"+*Ho(R8Fjw$\"+F4eUYF57%$\"+[ZhC7Fjw$\"+'*H'3V\"Fjw$\"+j4='=& F57%$\"+c$e**H\"Fjw$\"+><]E:Fjw$\"+[7AvdF57%$\"+E5bx8Fjw$\"+k@#oi\"Fjw $\"+i(y5T'F57%$\"+fFRd9Fjw$\"+I?'>t\"Fjw$\"+\"[I^4(F57%$\"+bN[R:Fjw$\" +9!f?%=Fjw$\"+#Q`(GyF57%$\"+9M#Qi\"Fjw$\"+:3Dd>Fjw$\"+VWK8')F57%$\"+NB T5Fjw$\"+H,srCFj w$\"+$)H')G7Fjw7%$\"+^'e#z?Fjw$\"+v9X9EFjw$\"+N'f[L\"Fjw7%$\"+))G4x@Fj w$\"+CQ5jFFjw$\"+*GxnW\"Fjw7%$\"+(=wrF#Fjw$\"+s[\"y\"HFjw$\"+WOvk:Fjw7 %$\"+\\&3&zBFjw$\"+=BsyIFjw$\"+'RE*)o\"Fjw7%$\"+u**3%[#Fjw$\"+gQ'fC$Fj w$\"+WKV>=Fjw7%$\"+i/#4f#Fjw$\"+&>x'>MFjw$\"+&)=Tc>Fjw7%$\"+8+++FFjw$ \"+@+++OFjw$\"+<+++@Fjw-%'COLOURG6&%$RGBGF(F($\"*++++\"Fjw-%(SCALINGG6 #%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%*THICKNESSG6#F+-%+PROJECTIONG6%$ !#IF)$\"#!)F)\"\"\"" 1 2 0 1 10 3 2 1 1 2 1 1.000000 80.000000 -30.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The t angent vector (or velocity) along the curve is" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "v:=D(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"vG7%R6#%\"tG6\"6$%)operatorG%&arrowGF),$9$\"\"'F)F)F)R6#%\"tGF)F*F), &*$)F.\"\"#\"\"\"\"\"$F8F7F)F)F)R6#%\"tGF)F*F),&F4F8F8!\"\"F)F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "and its length is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(t, real); lv:=len(v(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lvG,$*&-%%sqrtG6#\"\"#\"\"\",&F+F+*$)%#t| irGF*F+F+F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Along the cur ve, the temperature is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "T r:=T(op(r(t))); Tr:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #TrG,&*&,&*$)%#t|irG\"\"$\"\"\"F,*&F+F,F*F,F,F,)F*\"\"#F,F+*(F+F,F.F,, (F(F,*&F+F,F*F,!\"\"F+F,F,F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TrG ,&*$)%#t|irG\"\"$\"\"\"\"#=*&\"\"*F*)F(\"\"#F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The average value of the temperature is " } {XPPEDIT 18 0 "T[ave] = Int(T,s)/Int(1,s);" "6#/&%\"TG6#%$aveG*&-%$Int G6$F%%\"sG\"\"\"-F*6$F-F,!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int( T(r)*abs(v),t = 0 .. 3)/Int(abs(v),t = 0 .. 3);" "6#*&-%$IntG6$*&-%\"T G6#%\"rG\"\"\"-%$absG6#%\"vGF,/%\"tG;\"\"!\"\"$F,-F%6$-F.6#F0/F2;F4F5! \"\"" }{TEXT -1 6 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Note : The denominator is the arc length." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We compute the average temperature:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Tave:=Int(Tr*lv,t=0..3)/Int (lv,t=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TaveG*&-%$IntG6$,$* (,&*$)%#t|irG\"\"$\"\"\"\"#=*&\"\"*F0)F.\"\"#F0!\"\"F0-%%sqrtG6#F5F0,& F0F0*$F4F0F0F0F//F.;\"\"!F/F0-F'6$,$*&F7F0F:F0F/F " 0 "" {MPLTEXT 1 0 25 "Tave:=value(%); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TaveG#\"%xn\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++D%p\"!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "This can be compared to the average value in the box:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "1/(27*36*21)*Muint(T(x,y,z),x=0..27, y=0..36, z= 0..21); value(%); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$In tG6$-F%6$-F%6$,&*&%\"xG\"\"\"%\"yGF.F.*&F-F.%\"zGF.!\"\"/F-;\"\"!\"#F/ F/;F5\"#O/F1;F5\"#@#F.\"&7/#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$0% \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++]75!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "So the probe measures a temperature which is somewhat too high." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Another way to compute these line integrals is to us e the " }{TEXT 264 15 "Line_int_scalar" }{TEXT -1 23 " command (or its alias " }{TEXT 265 3 "Lis" }{TEXT -1 11 ") from the " }{TEXT 266 8 "v ec_calc" }{TEXT -1 82 " package which works directly with the parametr ized curve and the scalar function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Lis(T,r,t=0..3)/Lis(1,r,t=0..3); Tave:=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$IntG6$,$*(,&*$)%#t|irG\"\"$\"\"\" \"#=*&\"\"*F.)F,\"\"#F.!\"\"F.-%%sqrtG6#F3F.,&F.F.*$F2F.F.F.F-/F,;\"\" !F-F.-F%6$,$*&F5F.F8F.F-F:F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Tav eG#\"%xn\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 263 37 "The Mass and Center of Mass of a Wire " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "A wire initiall y has the shape of the parametric curve " }{XPPEDIT 18 0 "r(t) = [t*co s(t), t*sin(t), -abs(t)];" "6#/-%\"rG6#%\"tG7%*&F'\"\"\"-%$cosG6#F'F** &F'F*-%$sinG6#F'F*,$-%$absG6#F'!\"\"" }{TEXT -1 33 " where distances a re measured in " }{XPPEDIT 18 0 "cm;" "6#%#cmG" }{TEXT -1 56 ". Howev er, it has non-uniform thickness. At the point " }{XPPEDIT 18 0 "r(t) ;" "6#-%\"rG6#%\"tG" }{TEXT -1 42 " the cross sections are circles of \+ radius " }{XPPEDIT 18 0 "R(t) = .12+.1*cos(t);" "6#/-%\"RG6#%\"tG,&$\" #7!\"#\"\"\"*&$F,!\"\"F,-%$cosG6#F'F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "cm;" "6#%#cmG" }{TEXT -1 57 ". Assuming the wire is made from meta l whose density is " }{XPPEDIT 18 0 "rho = 1.1;" "6#/%$rhoG$\"#6!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "gm/(cm^3);" "6#*&%#gmG\"\"\"*$%#cmG \"\"$!\"\"" }{TEXT -1 64 ", we first want to find the mass and center \+ of mass of the wire." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 42 "If the wire is hung from the center point " }{XPPEDIT 18 0 "r(0) = [0, 0, 0];" "6#/-%\"rG6#\"\"!7%F'F'F'" }{TEXT -1 210 ", i t will rotate until the center of mass is directly below the suspensio n point. We want to plot the curve both before and after making this \+ rotation and then make an animation of the wire spinning about the " } {XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 12 "-axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We first input the p arametrized curve and plot it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:=[t*cos(t),t*sin(t),-abs(t)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%*&%#t|irG\"\"\"-%$cosG6#F'F(*&F'F(-%$sinGF+F(,$-%$absGF+ !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "spacecurve(r, t=- 2*Pi..2*Pi, orientation=[60,75], color=red, axes=normal, view=[default ,default,-2*Pi..0.1]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%'CURVESG6$7T7%$!+2`=$G'!\"*$!+tjIgS!#BF(7%$!+uPiHeF*$ !+xtqG:F*$!+k(Gn-'F*7%$!+9huF]F*$!+eQdJGF*$!+@AFqdF*7%$!+5f%3'RF*$!+`` 'e$QF*$!+yc\"Q^&F*7%$!+1fPDFF*$!+;Ez&\\%F*$!+N\"ftD&F*7%$!+[Z*GU\"F*$! +]X?%z%F*$!+#f-4+&F*7%$!+S)p1_\"!#5$!+H%3?u%F*$!+\\gWWZF*7%$\"+Ehr')** FT$!+SiYvVF*$!+1&*)z[%F*7%$\"+=gCd>F*$!+#>u;v$F*$!+jH`JUF*7%$\"+8uWsEF *$!+9;mUHF*$!+?k2vRF*7%$\"+$3Jl6$F*$!+JpjG?F*$!+x)>'=PF*7%$\"+3gy&G$F* $!+Ch&44\"F*$!+ML;iMF*7%$\"+A.7*>$F*$!+XM!R0#FT$!+\"z1d?$F*7%$\"+)ej[* GF*$\"+!GYxj&FT$!+[-D\\HF*7%$\"+)HBhU#F*$\"+(Rf$o6F*$!+0Pz#p#F*7%$\"+* 3R^&=F*$\"+DHIz:F*$!+irLOCF*7%$\"+Bg9Z7F*$\"+*fxyy\"F*$!+>1))z@F*7%$\" +]Y$Gk'FT$\"+cF20=F*$!+wSUB>F*7%$\"+3Fn+;FT$\"+kYEf;F*$!+Lv'pm\"F*7%$! +AS<^AFT$\"+b2V#R\"F*$!+!*4^59F*7%$!+U,UrYFT$\"+b>Gb5F*$!+ZW0a6F*7%$!+ s8V'f&FT$\"+:Iq%>Q^FT$\"+4StMQFT$!+3OT6kFT7%$!+ M!3dc$FT$\"+/_fV9FT$!+x\"[o%QFT7%$!+A_vr7FT$\"+>ouR;!#6$!+YFG#G\"FT7%$ \"+i^vr7FT$\"+jmuR;F_v$!+&o#G#G\"FT7%$\"+')zqlNFT$\"+g^fV9FT$!+;\"[o%Q FT7%$\"+kT>Q^FT$\"+TRtMQFT$!+ZNT6kFT7%$\"+x8V'f&FT$\"+LHqGb5F*$!+TW0a6F*7%$\"+'4u6D#FT$\"+[2V#R\"F*$!+% )4^59F*7%$!+.En+;FT$\"+fYEf;F*$!+Fv'pm\"F*7%$!+@X$Gk'FT$\"+aF20=F*$!+q SUB>F*7%$!+4g9Z7F*$\"+-w(yy\"F*$!+81))z@F*7%$!+u!R^&=F*$\"+KHIz:F*$!+c rLOCF*7%$!+'GBhU#F*$\"+4%f$o6F*$!+*p$z#p#F*7%$!+zN'[*GF*$\"+UkuPcFT$!+ U-D\\HF*7%$!+<.7*>$F*$!+\\K!R0#FT$!+&y1d?$F*7%$!+3gy&G$F*$!+-h&44\"F*$ !+GL;iMF*7%$!+!4Jl6$F*$!+4pjG?F*$!+r)>'=PF*7%$!+EuWsEF*$!+$fhE%HF*$!+9 k2vRF*7%$!+QgCd>F*$!+vTn^PF*$!+dH`JUF*7%$!+wjr')**FT$!+GiYvVF*$!++&*)z [%F*7%$\"+a&p1_\"FT$!+C%3?u%F*$!+VgWWZF*7%$\"+=Z*GU\"F*$!+`X?%z%F*$!+' e-4+&F*7%$\"+wePDFF*$!+FEz&\\%F*$!+H\"ftD&F*7%$\"+$)e%3'RF*$!+t`'e$QF* $!+sc\"Q^&F*7%$\"+#4Yx-&F*$!+&)QdJGF*$!+:AFqdF*7%$\"+fPiHeF*$!+6uqG:F* $!+e(Gn-'F*7%$\"+,`=$G'F*$!+O*[F)Q!#<$!+,`=$G'F*-%'COLOURG6&%$RGBG$\"* ++++\"!\")$\"\"!FjalFial-%*AXESSTYLEG6#%'NORMALG-%%VIEWG6%%(DEFAULTGFb bl;$!0ezrI&=$G'!#9$\"\"\"!\"\"-%+PROJECTIONG6%$\"#gFjal$\"#vFjalFhbl" 1 2 0 1 10 0 2 1 1 4 2 1.000000 75.000000 60.000000 1 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We can add the thickness of the wi re by switching to a tubeplot. 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$\"3-EVcn(4qP'F+$\"3lyF;x?OKbFD$!3WK\"RBNsU3'F+7%$\"3aEH&\\(ekwkF+$!3b i!*G`)GX7%F.$!3kw$*eR\\*o='F+7%$\"3!Gm0Ejfd['F+$!3OJ,\"[-\"QsoF.$!3**4 L)*fGdMjF+7%$\"3X&oG?h'3+kF+$!3$[0F.$!3u[aA9(f-W'F+7%F`aw$\"3S&f)3HNL^eF.Fdaw-%&COLORG6&%$RGBG$\"*++++ \"!\")$\"\"!FiewFhew-%*AXESSTYLEG6#%'NORMALG-%+PROJECTIONG6%$\"#gFiew$ \"#vFiew\"\"\"" 1 2 0 1 10 0 2 1 1 4 2 1.000000 75.000000 60.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Next, we find the \+ tangent vector (velocity) to the curve and its length:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v:=diff(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%,&-%$cosG6#%#t|irG\"\"\"*&F*F+-%$sinGF)F+!\"\", &F-F+*&F*F+F'F+F+,$-%'signumGF)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "lv:=len(v); simplify(%); lv2:=subs(abs(1,t)^2=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lvG*$-%%sqrtG6#,(*$),&-%$cosG6#%# t|irG\"\"\"*&F0F1-%$sinGF/F1!\"\"\"\"#F1F1*$),&F3F1*&F0F1F-F1F1F6F1F1* $)-%'signumGF/F6F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6# ,&*$)%#t|irG\"\"#\"\"\"F,F+F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ lv2G*$-%%sqrtG6#,&*$)%#t|irG\"\"#\"\"\"F.F-F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "At the point " }{XPPEDIT 18 0 "r(t);" "6#-%\"rG6#%\" tG" }{TEXT -1 30 " the cross sectional radius is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "R:=.12+.1*cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,&$\"#7!\"#\"\"\"*&$F)!\"\"F)-%$cosG6#%#t|irGF)F) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "and the density is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rho:=1.1;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$rhoG$\"#6!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "So the mass of a segment of length " }{XPPEDIT 18 0 "ds = abs(v )*dt;" "6#/%#dsG*&-%$absG6#%\"vG\"\"\"%#dtGF*" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "dm = rho*Pi*R^2*abs(v)*dt;" "6#/%#dmG*,%$rhoG\"\"\"%#Pi GF'%\"RG\"\"#-%$absG6#%\"vGF'%#dtGF'" }{TEXT -1 23 " and the total mas s is " }{XPPEDIT 18 0 "Int(rho*Pi*R^2*abs(v),t = -2*Pi .. 2*Pi);" "6#- %$IntG6$**%$rhoG\"\"\"%#PiGF(%\"RG\"\"#-%$absG6#%\"vGF(/%\"tG;,$*&F+F( F)F(!\"\"*&F+F(F)F(" }{TEXT -1 17 ". We compute: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Int(rho*Pi*R^2*lv2,t=-2*Pi..2*Pi); value( %); M:=evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(%#PiG \"\"\"),&$\"#7!\"#F)*&$F)!\"\"F)-%$cosG6#%#t|irGF)F)\"\"#F)-%%sqrtG6#, &*$)F5F6F)F)F6F)F)$\"#6F1/F5;,$F(F.,$F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$,$*(%#PiG\"\"\"),&$\"#7!\"#F)*&$F)!\"\"F)-%$co sG6#%#t|irGF)F)\"\"#F)-%%sqrtG6#,&*$)F5F6F)F)F6F)F)$\"#6F1/F5;,$F(F.,$ F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"+8=C.J!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The first moments are found by add ing factors of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "z;" "6 #%\"zG" }{TEXT -1 124 " (in terms of the curve) inside the integral. \+ The center of mass is then found by dividing the first moments by the \+ mass. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "mom:=map(Int,eva ll(rho*Pi*R^2*lv2*r),t=-2*Pi..2*Pi); mom:=evalf(%);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$momG7%-%$IntG6$,$*,%#PiG\"\"\"),&$\"#7!\"#F,*&$F,! \"\"F,-%$cosG6#%#t|irGF,F,\"\"#F,-%%sqrtG6#,&*$)F8F9F,F,F9F,F,F8F,F5F, $\"#6F4/F8;,$F+F1,$F+F9-F'6$,$*,F+F,F-F,F:F,F8F,-%$sinGF7F,F@FB-F'6$,$ **F+F,F-F,F:F,-%$absGF7F,$!#6F4FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$momG7%$\"\"!F'$!+dja[l!\"*$!+8Cs#R\"!\")" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "cm:=mom/M;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cm G7%$\"\"!F'$!+YvA5@!\"*$!+z'fz[%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Notice that the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 36 "- component of the center of mass is " }{XPPEDIT 18 0 "0;" "6#\"\"!" } {TEXT -1 37 " as should be expected from symmetry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "If the wire is hung from \+ " }{XPPEDIT 18 0 "[0, 0, 0];" "6#7%\"\"!F$F$" }{TEXT -1 26 " it will r otate about the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 18 "-axis by an angle " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "phi:=arctan(c m[2]/cm[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG$\"+PmA&R%!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The rotation matrix is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "T:=matrix([[1,0,0],[0,cos(ph i),-sin(phi)],[0,sin(phi),cos(phi)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+$\"+l([&\\!*!#5$!+da2bUF/ 7%F+$\"+da2bUF/F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "So the rotat ed curve is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r2:=evalm(T \+ &* r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G-%'vectorG6#7%*&%#t|ir G\"\"\"-%$cosG6#F*F+,&*&F*F+-%$sinGF.F+$\"+l([&\\!*!#5*&$\"+da2bUF5F+- %$absGF.F+F+,&F0F7*&$\"+l([&\\!*F5F+F9F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "To check this, we also rotate the center of mass" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cm2:=evalm(T &* cm);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cm2G-%'vectorG6#7%$\"\"!F*F)$!+T(=$ f\\!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The center of mass is \+ now on the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 73 "-axis. (The f irst time I did this, I got the rotation matrix transposed.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "We plot the new curve:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "tubeplot(r2, rad ius=R, t=-2*Pi..2*Pi, orientation=[60,75], color=red, axes=normal);" } }{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%MESHG6$7T7,7 %$!3)\\l?jwQ?2'!#<$\"3U8&*)o`KIt#F+$!3khE49FKpcF+7%$!3U13(GnP97'F+$\"3 G\\yt#HW4o#F+$!30Io&o')fq`&F+7%$!3S^(Q^4?lC'F+$\"3_.>!))y#RDEF+$!31y6C \"Q([uaF+7%$!3_U!4wde()Q'F+$\"3k+&f6>rBf#F+$!3?')4o=O)3^&F+7%$!3C#=HN: )f\"['F+$\"3c*pTBnItf#F+$!3*e/z^P=#HcF+7%FF$\"3Yb[)fj]zj#F+$!3C*eDI]@T x&F+7%F?$\"3CE(f'\\XA&p#F+$!3_NBG=7zxeF+7%F8$\"3e!>HEJ`Bu#F+$!3K3,-4$> <*eF+7%F1$\"3W!)>#H&[GdFF+$!3&p@*ym')Q4eF+F(7,7%$!3w\\pm$*o(4j&F+$\"3G *3\\yD#ek7F+$!3'f4uKL_93'F+7%$!3GhroT&Qyn&F+$\"3E&*ou'\\3!47F+$!3+9HEY mC_fF+7%$!3waq)p=Adz&F+$\"3=7H)ppK.9\"F+$!31?ksZCB%*eF+7%$!3NG+!=wo%Hf F+$\"3#\\?8&G!*o!4\"F+$!3]Qe\"fAbX$fF+7%$!35$[Vtv'\\;gF+$\"3UnEa:jI$3 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