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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 37 "SAMPLE MAP LE WORKSHEET FOR THE TORUS:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 44 "First, we'll get the equations of the torus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=cos(beta)*(cos(alpha)+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fG*&-%$cosG6#%%betaG\"\"\",&-F'6#%&alphaGF*\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:=sin(beta)*(cos(alpha)+2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&-%$sinG6#%%betaG\"\"\",&-%$cos G6#%&alphaGF*\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h :=sin(alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG-%$sinG6#%&alph aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F1:=[f,g,h];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F1G7%*&-%$cosG6#%%betaG\"\"\",&-F(6 #%&alphaGF+\"\"#F+F+*&-%$sinGF)F+F,F+-F3F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 89 "plot3d(F1,alpha=0..2*Pi,beta=0..2*Pi,style=patchnog 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of time:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F:=subs(\{alpha=-t*(t-Pi),beta=t\},F1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%*&-%$cosG6#%\"tG\"\"\",&-F(6#, $*&F*F+,&F*F+%#PiG!\"\"F+F3F+\"\"#F+F+*&-%$sinGF)F+F,F+-F7F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 258 68 "The next few lines give a numeric al approximation of the arc length:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dF:=diff(F,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#dFG7%,&*&-%$sinG6#%\"tG\"\"\",&-%$cosG6#* &F+F,,&F+F,%#PiG!\"\"F,F,\"\"#F,F,F4*(-F/F*F,-F)F0F,,&*&F5F,F+F,F,F3F4 F,F4,&*&F7F,F-F,F,*(F(F,F8F,F9F,F4,$*&F.F,F9F,F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L:=sqrt(dF[1]^2+dF[2]^2+dF[3]^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"LG*$,(*$),&*&-%$sinG6#%\"tG\"\"\",&-%$co sG6#*&F.F/,&F.F/%#PiG!\"\"F/F/\"\"#F/F/F7*(-F2F-F/-F,F3F/,&*&F8F/F.F/F /F6F7F/F7F8F/F/*$),&*&F:F/F0F/F/*(F+F/F;F/F " 0 "" {MPLTEXT 1 0 28 "ArcL:=evalf(Int( L,t=0..Pi));" 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