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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 April 2
0" }}{PARA 256 "" 0 "" {TEXT -1 47 "Christoffel symbols and Gauss' The
orem Egregium" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "wit
h(linalg):\n  #as much as possible we will use matrix algebra" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce 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:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nrm:=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 0 206 "#Derivative matrix for map
ping 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),di
ff(X[3],v)]);\nsimplify(augment(Xu,Xv));\nend:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 113 "#Matrix of first fundamental form:\ngij:=proc
(X)\nlocal Y;\nY:=evalm(DXq(X));\nsimplify(evalm(transpose(Y)&*Y));\ne
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "#unit normal:\nN:=p
roc(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 0 382 "#matrix of second fundamental form:\nhij:=proc(X)\nl
ocal Y,Xu,Xv,Xuu,Xuv,Xvv,U,h11,h12,h22;\nY:=DXq(X);\nU:=N(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,U);\nh12:=dp(Xuv,U);\n
h22:=dp(Xvv,U);\nsimplify(matrix(2,2,[h11,h12,h12,h22]));\nend:\n" }}}
{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;\n
H:=hij(X);\nG:=gij(X);\nsimplify(evalm(inverse(G)&*H));\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "#Gauss curvature from second
 fundamental form\nGK2:=proc(X)\nlocal A;\nA:=aij(X);\ndet(A);\nend:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "#Mean curvature\nMK:=proc
(X)\nlocal A;\nA:=aij(X);\n(1/2)*trace(A);\nend:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 621 "#Christoffel symbols\nGamma:=proc(X)\nlocal \+
Gammas,G,Ginv,DG,i,j,k,l;\nG:=X;\nGinv:=evalm(inverse(G));\nDG:=array(
1..2,1..2,1..2);\n   DG[1,1,1]:=diff(G[1,1],u);\n   DG[1,1,2]:=diff(G[
1,1],v);\n   DG[1,2,1]:=diff(G[1,2],u);\n   DG[2,1,1]:=DG[1,2,1];\n   \+
DG[1,2,2]:=diff(G[1,2],v);\n   DG[2,1,2]:=DG[1,2,2];\n   DG[2,2,1]:=di
ff(G[2,2],u);\n   DG[2,2,2]:=diff(G[2,2],v);\nGammas:=array(1..2,1..2,
1..2);\n   for i from 1 to 2 do\n      for j from 1 to 2 do\n         \+
for l from 1 to 2 do\n         Gammas[i,j,l]:=sum((1/2)*Ginv[k,l]*\n  \+
         (DG[i,k,j]+DG[k,j,i]-DG[i,j,k]),k=1..2);\n         od:\n     \+
  od:\n     od:\nsimplify(Gammas);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 354 "#derivatives of Christoffel symbols\nDGamma:=proc(
X)\nlocal G,Gammas,DGammas,i,j,k,m;\nG:=X;\nGammas:=Gamma(G);\nDGammas
:=array(1..2,1..2,1..2,1..2);\nfor i from 1 to 2 do\n  for j from 1 to
 2 do\n    for m from 1 to 2 do\n      DGammas[i,j,m,1]:=diff(Gammas[i
,j,m],u);\n      DGammas[i,j,m,2]:=diff(Gammas[i,j,m],v);\n    od:\n  \+
 od:\n  od:\nsimplify(DGammas);\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 361 "#Gauss curvature from first fundamental form\nGK1:
=proc(X)\nlocal G,Gammas,DGammas,Ginv,K,i,r;\nG:=X;\nGinv:=inverse(G);
\nGammas:=Gamma(G);\nDGammas:=DGamma(G);\n   \n  K:=sum(Ginv[1,i]*(DGa
mmas[i,1,2,2] - DGammas[i,2,2,1]\n        + sum(\n           Gammas[i,
1,r]*Gammas[r,2,2] - Gammas[i,2,r]*Gammas[r,1,2],\n          r=1..2)),
\n       i=1..2);\n   \n  \nsimplify(K);\nend:\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "test:=(u,v)->[u,v,k1*u^
2+k2*v^2];\n  #good idea to check our derivation on a graph:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%testGR6$%\"uG%\"vG6\"6$%)operatorG%&arrow
GF)7%9$9%,&*&%#k1G\"\"\")F.\"\"#F3F3*&%#k2GF3)F/F5F3F3F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "GK1(gij(test(u,v)));\n  #fro
m the first fundamental form" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&
%#k1G\"\"\"%#k2GF'F'*$),(F'F'*(\"\"%F')F(\"\"#F')%\"vGF/F'F'*(F-F')F&F
/F')%\"uGF/F'F'F/F'!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "GK2(test(u,v));\n  #from the differential of the normal map" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%#k1G\"\"\"%#k2GF'F'*$),(F'F'*(
\"\"%F')F(\"\"#F')%\"vGF/F'F'*(F-F')F&F/F')%\"uGF/F'F'F/F'!\"\"F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "stinv:=(u,v)->[2*u/(u^2+v^2
+1),2*v/(u^2+v^2+1),(u^2+v^2-1)/(u^2+v^2+1)];\n   #inverse of stereogr
aphic projection onto the sphere" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&stinvGR6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)7%,$*&9$\"\"\",(*$)F0\"
\"#F1F1*$)9%F5F1F1F1F1!\"\"F5,$*&F8F1F2F9F5*&,(F3F1F6F1F1F9F1F2F9F)F)F
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "GK1(gij(stinv(u,v)));
\nGK2(stinv(u,v));\n   #notice the second expression also equals 1; it
 would've been\n   #neater if I'd assumed v,u real." }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*$)-%%csgnG
6#*&,2*&)-%*conjugateG6#,&*$)%\"uG\"\"#\"\"\"F4*$)%\"vGF3F4F4F3F4)F2\"
\")F4F4**\"\"%F4F+F4)F2\"\"'F4F6F4F4**F=F4F+F4)F2F;F4)F7F;F4F4**F;F4F+
F4F1F4)F7F=F4F4*&F+F4)F7F9F4F4*(F3F4)-%$absGF.F=F4F1F4F4*(F3F4FFF4F6F4
F4*$)FGF9F4F4F4F+F4F3F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 
"helcat:=(u,v)->[cos(t)*sinh(v)*sin(u)+sin(t)*cosh(v)*cos(u),\n  -cos(
t)*sinh(v)*cos(u)+sin(t)*cosh(v)*sin(u),u*cos(t)+v*sin(t)];\n#in your \+
homework you showed these are all isometric as the parameter\n#t varie
s, so Gauss curvature should be t-independent" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'helcatGR6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)7%,&*(
-%$cosG6#%\"tG\"\"\"-%%sinhG6#9%F4-%$sinG6#9$F4F4*(-F:F2F4-%%coshGF7F4
-F1F;F4F4,&*(F0F4F5F4FAF4!\"\"*(F>F4F?F4F9F4F4,&*&F<F4F0F4F4*&F8F4F>F4
F4F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "GK1(gij(helcat(
u,v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$)-%%coshG6#%
\"vG\"\"%F%!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "GK2(h
elcat(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)-%%csgnG6#*$)-%
%coshG6#%\"vG\"\"#\"\"\"F0F1,&*$)-%$sinG6#%\"tGF0F1F1*$)-%$cosGF7F0F1F
1F1F1*$)F,\"\"%F1!\"\"F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
221 "hypgij:=(u,v)->matrix(2,2,[1/v^2,0,0,1/v^2]);\n#the hyperbolic pl
ane is the upper half plane, with conformal metric\n#in which g11=g22=
1/v^2.  This \"surface\" is not realizable as a surface\n#in R^3, yet \+
it has a \"curvature\":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'hypgijGR
6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)-%'matrixG6%\"\"#F07&*&\"\"\"F3*
$)9%F0F3!\"\"\"\"!F8F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "GK1(hypgij(u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26 0 0" 221 }
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