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{SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 9 "Math 4530" }}{PARA 258 "" 0 "" {TEXT -1 15 "Friday March 2
8" }}{PARA 256 "" 0 "" {TEXT -1 43 "first and second fundamental forms
 in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
420 "Here is a list of procedures to calculate first and second fundam
ental forms, as well as Gauss and Mean curvature, for regular paramete
rized surfaces.  The procedures are modified from ones given on pages \+
119-121 of Oprea, mainly to use matrix algebra as we have done in clas
s.  They are illustrated with computations for the Enneper Surface, wh
ich was a hw problem from last week which was somewhat messy to do by \+
hand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "restart:\n  #always a good idea" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 63 "with(linalg):\n  #as much as possible we will
 use matrix algebra" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the prote
cted names norm and trace have been redefined and unprotected\n" }}}
{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:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dp([1,2,3],[-1,2
,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "#2-norm\nnrm:=proc(X)\nsqrt(dp(X,X));\nend:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "nrm([1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%
%sqrtG6#\"#9\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "#cro
ss product:\nxp:=proc(X,Y)\nlocal a,b,c;\na:=X[2]*Y[3]-X[3]*Y[2];\nb:=
X[3]*Y[1]-X[1]*Y[3];\nc:=X[1]*Y[2]-X[2]*Y[1];\n[a,b,c];\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xp([1,0,0],[0,2,0]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 206 "#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)]);\nsimp
lify(augment(Xu,Xv));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
100 "#Enneper surface parameterization, to play with:\nEnne:=(u,v)->[u
-u^3/3+u*v^2,v-v^3/3+v*u^2,u^2-v^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%EnneGR6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)7%,(9$\"\"\"*&#F0\"\"
$F0*$)F/F3F0F0!\"\"*&F/F0)9%\"\"#F0F0,(F9F0*&#F0F3F0*$)F9F3F0F0F6*&F9F
0)F/F:F0F0,&*$FAF0F0*$F8F0F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "DXq(Enne(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'matrixG6#7%7$,(\"\"\"F)*$)%\"uG\"\"#F)!\"\"*$)%\"vGF-F)F),$*&F1F)F,F)
F-7$F2,(F)F)F/F.F*F)7$,$F,F-,$F1!\"#" }}}{EXCHG {PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "#Matrix of
 first fundamental form:\ngij:=proc(X)\nlocal g11,g12,g22,Y;\nY:=evalm
(DXq(X));\nsimplify(evalm(transpose(Y)&*Y));\nend:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "gij(Enne(u,v));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$,.*&)%\"vG\"\"#\"\"\")%\"uGF,F-F,F-F-*&
F,F-F*F-F-*&F,F-F.F-F-*$)F+\"\"%F-F-*$)F/F4F-F-\"\"!7$F7F(" }}}{EXCHG 
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 116 "#unit normal:\nN:=proc(X)\nlocal Y,Z,s;\nY:=DXq(X);\nZ:=xp(col(
Y,1),col(Y,2));\ns:=nrm(Z);\nsimplify(evalm((1/s)*Z));\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "N(Enne(u,v));\n#if you look
 up csgn you see that it stands for\n#\"complex sign;\" for real numbe
rs it is +1 when\n#the expression inside is positive and -1 when it\n#
is negative." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&*&-%
%csgnG6#*$),(*$)%\"uG\"\"#\"\"\"F4F4F4*$)%\"vGF3F4F4F3F4F4F2F4F4F/!\"
\"!\"#,$*&*&F*F4F7F4F4F/F8F3,$*&*&F*F4,(F5F4F0F4F4F8F4F4F/F8F8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "#matrix of second fundament
al form:\nhij:=proc(X)\nlocal Y,Xu,Xv,Xuu,Xuv,Xvv,U,h11,h12,h22;\nY:=D
Xq(X);\nU:=N(X);\nXu:=col(Y,1);\nXv:=col(Y,2);\nXuu:=[diff(Xu[1],u),di
ff(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(X
uu,U);\nh12:=dp(Xuv,U);\nh22:=dp(Xvv,U);\nsimplify(matrix(2,2,[h11,h12
,h12,h22]));\nend:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "hij(Enne(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7
$,$-%%csgnG6#*$),(*$)%\"uG\"\"#\"\"\"F3F3F3*$)%\"vGF2F3F3F2F3F2\"\"!7$
F7,$F)!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "#matrix A of the (opposite) of the
 differential\n#of the normal map\naij:=proc(X)\nlocal Y,H,G;\nH:=hij(
X);\nG:=gij(X);\nsimplify(evalm(inverse(G)&*H));\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "aij(Enne(u,v));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7$7$,$*&-%%csgnG6#*$),(*$)%\"uG\"\"#\"\"
\"F4F4F4*$)%\"vGF3F4F4F3F4F4,.*&F6F4F1F4F3F4F4*&F3F4F6F4F4*&F3F4F1F4F4
*$)F7\"\"%F4F4*$)F2F>F4F4!\"\"F3\"\"!7$FB,$F)!\"#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 61 "#Gauss curvature\nGK:=proc(X)\nlocal A;\nA:=
aij(X);\ndet(A);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#
Mean curvature\nMK:=proc(X)\nlocal A;\nA:=aij(X);\ntrace(A);\nend:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "GK(Enne(u,v));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*&*$)-%%csgnG6#*$),(*$)%\"uG\"\"#\"\"\"F1F1F1
*$)%\"vGF0F1F1F0F1F0F1F1*$),.*&F3F1F.F1F0F1F1*&F0F1F3F1F1*&F0F1F.F1F1*
$)F4\"\"%F1F1*$)F/F=F1F1F0F1!\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "MK(Enne(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 5 0" 
420 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }