{VERSION 5 0 "IBM INTEL NT" "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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 
1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 
0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 
0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 311 "The following Maple Proce
dure, called Curvature, accepts a vector valued function and outputs t
he standard expressions E, F, G, e, f, and g followed by the Gaussian \+
and Mean Curvatures.  More specifically, the vector valued function sh
ould be a parameterization of a surface and so a function from the pla
ne to " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 166 ", and fu
rthermore, the function must be entered with u and v as the variables.
  To ensure this works, you must hit enter to make sure maple executes
 the lines below. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 908 "with
(LinearAlgebra):\nCurvature:=proc(x)\nlocal xu,xv,E,F,G,e,f,g,xuu,xvv,
xuv,N,k,mk;\nxu:=<diff(x[1],u),diff(x[2],u),diff(x[3],u)>;\nxv:=<diff(
x[1],v),diff(x[2],v),diff(x[3],v)>;\nE:=DotProduct(xu,xu,conjugate=fal
se);\nF:=DotProduct(xu,xv,conjugate=false);\nG:=DotProduct(xv,xv,conju
gate=false);\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)>;\nN:=CrossProduct(xu,xv)/sqrt(E*G-F^2);\ne:=D
otProduct(N,xuu,conjugate=false);\nf:=DotProduct(N,xuv,conjugate=false
);\ng:=DotProduct(N,xvv,conjugate=false);\nk:=(e*g-f^2)/(E*G-F^2);\nmk
:=(G*e+E*g-2*F*f)/(2*E*G-2*F^2);\nprintf(\"%s\",\"E, F, G = \");\nprin
t(simplify(E),simplify(F),simplify(G));\nprintf(\"%s\",\"e, f, g = \")
;\nprint(simplify(e),simplify(f),simplify(g));\nprintf(\"%s\",\"Gaussi
an Curvature =\");\nprint(simplify(k));\nprintf(\"%s\",\"Mean Curvatur
e =\");\nprint(simplify(mk));\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 59 "Now all one has to do is enter the function.  For example  " }
{XPPEDIT 18 0 "x = (u*cos(v), u*sin(v), 0);" "6#/%\"xG6%*&%\"uG\"\"\"-
%$cosG6#%\"vGF(*&F'F(-%$sinG6#F,F(\"\"!" }{TEXT -1 126 ", a non-standa
rd parametrization of the plane; simply enter \"Curvature(<u*cos(v),u*
sin(v),0>);\" and hit enter, as shown below." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "Curvature(<u*cos(v),u*sin(v),0>);" }}{PARA 6 "" 
1 "" {TEXT -1 10 "E, F, G = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"
\"\"!*$)%\"uG\"\"#F#" }}{PARA 6 "" 1 "" {TEXT -1 10 "e, f, g = " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#F#" }}{PARA 6 "" 1 "" {TEXT -1 
20 "Gaussian Curvature =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
{PARA 6 "" 1 "" {TEXT -1 16 "Mean Curvature =" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "And one c
an easily check the curvature of the standard torus:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "Curvature(<(R+r*cos(u))*cos(v),(R+r*cos(u
))*sin(v),r*sin(u)>);" }}{PARA 6 "" 1 "" {TEXT -1 10 "E, F, G = " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%*$)%\"rG\"\"#\"\"\"\"\"!,(*$)%\"RGF&F'
F'**F&F'F,F'F%F'-%$cosG6#%\"uGF'F'*&F$F')F.F&F'F'" }}{PARA 6 "" 1 "" 
{TEXT -1 10 "e, f, g = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*(%\"rG\"\"
#,&%\"RG\"\"\"*&F$F(-%$cosG6#%\"uGF(F(F(*&)F$F%F()F&F%F(#!\"\"F%\"\"!*
*F$F(F*F(F&F%F.F1" }}{PARA 6 "" 1 "" {TEXT -1 20 "Gaussian Curvature =
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$cosG6#%\"uG\"\"\",&%\"RGF(*&%
\"rGF(F$F(F(!\"\"F,F-" }}{PARA 6 "" 1 "" {TEXT -1 16 "Mean Curvature =
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"RG\"\"\"*(\"\"#F'%\"rGF'-
%$cosG6#%\"uGF'F'F'*&)F*F)F'),&F&F'*&F*F'F+F'F'F)F'#!\"\"F)#F'F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }