{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 }{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 "" 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 15 "Monday March 28" }}
{PARA 256 "" 0 "" {TEXT -1 42 "Computations related to the shape opera
tor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "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 illustrated with comput
ations and pictures for the helicoid and the torus." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:\n
with(linalg):\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
74 "assume(u,real); #this gets rid of that annoying \"csgn\" fcn\nassu
me(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:=proc(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)]);\nsimplify(augment(Xu,Xv),radical,symbolic,trig);\nen
d:" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "#Matrix of f
irst fundamental form:\ngij:=proc(X)\nlocal g11,g12,g22,Y;\nY:=evalm(D
Xq(X));\nsimplify(evalm(transpose(Y)&*Y),\n   radical,symbolic,trig);
\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "#unit 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 fundament
al 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)];\nh11:=dp
(Xuu,U1);\nh12:=dp(Xuv,U1);\nh22:=dp(Xvv,U1);\nsimplify(matrix(2,2,[h1
1,h12,h12,h22]),\n  radical,symbolic,trig);\nend:\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 155 "#matrix of shape operator wrt basis \{Xu
,Xv\}:\naij:=proc(X)\nlocal Y,H,G;\nH:=hij(X);\nG:=gij(X);\nsimplify(e
valm(inverse(G)&*H),\n   radical,symbolic,trig);\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "#Gauss curvature\nGK:=proc(X)\nloca
l A;\nA:=aij(X);\nsimplify(det(A),radical,symbolic,trig);\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "#Mean curvature\nMK:=proc(X)
\nlocal A;\nA:=aij(X);\nsimplify(1/2*trace(A),radical,symbolic,trig);
\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "#Principle curvat
ures and directions:\nPK:=proc(X)\nlocal Y;\nY:=aij(X);\neigenvects(Y)
;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "test:=[u,v,u^2-v
^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "DXq(test);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "gij(test);\nhij(test);\nsub
s(\{u=0,v=0\},aij(test));\nsubs(\{u=0,v=0\},GK(test));\nsubs(\{u=0,v=0
\},MK(test));\nsubs(\{u=0,v=0\},aij(test));\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "torus:=[(2+cos(u))*cos(v),(2+cos(u))*sin(v),sin(
u)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gij(torus);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "hij(torus);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "aij(torus);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "GK(torus);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "MK(torus);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "aij(torus);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(t
orus,u=0..2*Pi,v=0..2*Pi,color=GK(torus));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "hel:=[v*cos(u),v*sin(u),u];\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 56 "gij(hel);\nhij(hel);\naij(hel);\nPK(hel);\nMK
(hel);\nGK(hel);" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"plot3d(hel,u=0..2*Pi,v=-3..3,color=GK(hel));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{MARK "38" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }