{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 258 "" 0 "" {TEXT -1 14 "Monday March 3" }} {PARA 256 "" 0 "" {TEXT -1 42 "Computations related to the shape opera tor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 381 "H ere is a list of procedures to calculate the matrix of the shape opera tor, the principle curvatures, the mean curvature and the Gauss curvat ure, using a given patch X. The procedures are modified from ones giv en on pages 119-121 of Oprea, using notation and linear algebra from c lass. The procedures are illustrated with computations and pictures f or the helicoid and the torus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:\nwith(linalg):\nwit h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "assume(u,real) ; #this gets rid of that annoying \"csgn\" fcn\nassume(v,real);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#dot product\ndp:=proc(X,Y) \nX[1]*Y[1]+X[2]*Y[2]+X[3]*Y[3];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#2-norm, i.e. magnitude.\nnrm:=proc(X)\nsqrt(dp(X,X)) ;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "#cross product: \nxp:=proc(X,Y)\nlocal a,b,c;\na:=X[2]*Y[3]-X[3]*Y[2];\nb:=X[3]*Y[1]-X [1]*Y[3];\nc:=X[1]*Y[2]-X[2]*Y[1];\n[a,b,c];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "#Derivative matrix for mapping X:\nDXq:=pr oc(X)\nlocal Xu,Xv;\nXu:=matrix(3,1,[diff(X[1],u),diff(X[2],u),diff(X[ 3],u)]);\nXv:=matrix(3,1,[diff(X[1],v),diff(X[2],v),diff(X[3],v)]);\ns implify(augment(Xu,Xv),radical,symbolic,trig);\nend:" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "#Matrix of first fundamental form:\ngij:=proc(X)\nlocal g11,g12,g22,Y ;\nY:=evalm(DXq(X));\nsimplify(evalm(transpose(Y)&*Y),\n radical,sym bolic,trig);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "#uni t normal:\nU:=proc(X)\nlocal Y,Z,s;\nY:=DXq(X);\nZ:=xp(col(Y,1),col(Y, 2));\ns:=nrm(Z);\nsimplify(evalm((1/s)*Z),radical,symbolic,trig);\nend :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "#matrix of second fun damental form:\nhij:=proc(X)\nlocal Y,Xu,Xv,Xuu,Xuv,Xvv,U1,h11,h12,h22 ;\nY:=DXq(X);\nU1:=U(X);\nXu:=col(Y,1);\nXv:=col(Y,2);\nXuu:=[diff(Xu[ 1],u),diff(Xu[2],u),diff(Xu[3],u)];\nXuv:=[diff(Xu[1],v),diff(Xu[2],v) ,diff(Xu[3],v)];\nXvv:=[diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)];\nh 11:=dp(Xuu,U1);\nh12:=dp(Xuv,U1);\nh22:=dp(Xvv,U1);\nsimplify(matrix(2 ,2,[h11,h12,h12,h22]),\n radical,symbolic,trig);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "#matrix of shape operator wrt basi s \{Xu,Xv\}:\naij:=proc(X)\nlocal Y,H,G;\nH:=hij(X);\nG:=gij(X);\nsimp lify(evalm(inverse(G)&*H),\n radical,symbolic,trig);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "#Gauss curvature\nGK:=proc(X )\nlocal A;\nA:=aij(X);\nsimplify(det(A),radical,symbolic,trig);\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "#Mean curvature\nMK:=pr oc(X)\nlocal A;\nA:=aij(X);\nsimplify(1/2*trace(A),radical,symbolic,tr ig);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "#Principle cu rvatures and directions:\nPK:=proc(X)\nlocal Y;\nY:=aij(X);\neigenvect s(Y);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "test:=[u,v,u ^2-v^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG7%%#u|irG%#v|irG,& *$)F&\"\"#\"\"\"F,*$)F'F+F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "DXq(test);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr ixG6#7%7$\"\"\"\"\"!7$F)F(7$,$%#u|irG\"\"#,$%#v|irG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "gij(test);\nhij(test);\nsubs(\{u=0 ,v=0\},aij(test));\nsubs(\{u=0,v=0\},GK(test));\nsubs(\{u=0,v=0\},MK(t est));\nsubs(\{u=0,v=0\},aij(test));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&\"\"\"F)*&\"\"%F))%#u|irG\"\"#F)F),$*&F-F)%#v| irGF)!\"%7$F/,&F)F)*&F+F))F1F.F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'matrixG6#7$7$,$*&\"\"\"F**$-%%sqrtG6#,(*$)%#u|irG\"\"#F*\"\"%*&F4F* )%#v|irGF3F*F*F*F*F*!\"\"F3\"\"!7$F9,$F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"#\"\"!7$F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"#\"\"!7$F)!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "torus:=[(2+cos(u))*cos(v),(2 +cos(u))*sin(v),sin(u)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&torusG7 %*&,&\"\"#\"\"\"-%$cosG6#%#u|irGF)F)-F+6#%#v|irGF)*&F'F)-%$sinGF/F)-F3 F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gij(torus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F),(\"\"%F(*&F, F(-%$cosG6#%#u|irGF(F(*$)F.\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "hij(torus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7$7$\"\"\"\"\"!7$F)*&-%$cosG6#%#u|irGF(,&\"\"#F(F,F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "aij(torus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F)*&-%$cosG6#%#u|irGF (,&\"\"#F(F,F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "GK(t orus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#%#u|irG\"\"\",&\" \"#F(F$F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "MK(torus) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$cosG6#%#u|irGF%F%, &\"\"#F%F&F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "aij(to rus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$ F)*&-%$cosG6#%#u|irGF(,&\"\"#F(F,F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(torus,u=0..2*Pi,v=0..2*Pi,color=GK(torus));" } }{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 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