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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "The following Maple Proce
dure, called FirstForm, accepts a vector valued function and outputs t
he matrix associated with the First Fundamental Form.  More specifical
ly, the vector valued function should be a parameterization of a surfa
ce and so a function from the plane to " }{XPPEDIT 18 0 "R^3;" "6#*$%
\"RG\"\"$" }{TEXT -1 172 ", and furthermore, the function must be ente
red with u and v as the variables.  To ensure this works, you must hit
 enter twice to make sure maple executes the lines below. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 347 "FirstForm:=proc(x)\nlocal xu,xv,E,
F,G,g;\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:=(u,v)->DotProduct(xu,xu,conjugate=f
alse);\nF:=(u,v)->DotProduct(xu,xv,conjugate=false);\nG:=(u,v)->DotPro
duct(xv,xv,conjugate=false);\ng:=array([[(E(u,v)),(F(u,v))],[(F(u,v)),
(G(u,v))]]);\nsimplify(g);\nprint(%);\nend:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 90 "Now all one has to do is enter the function.  For examp
le is x is a parameterization with " }{XPPEDIT 18 0 "x = (u*cos(v), u*
sin(v), 0);" "6#/%\"xG6%*&%\"uG\"\"\"-%$cosG6#%\"vGF(*&F'F(-%$sinG6#F,
F(\"\"!" }{TEXT -1 86 ", then simply enter \"FirstForm(<u*cos(v),u*sin
(v),0>);\" and hit enter, as shown below." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "FirstForm(<u*cos(v),u*sin(v),0>);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F)*$)%\"uG\"\"#F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "And one can easily check the answe
r to problem 1b, page 99, by doing the following:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "FirstForm(<a*u*cos(v),b*u*sin(v),u^2>);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,**&)%\"aG\"\"#\"\"\")
-%$cosG6#%\"vGF,F-F-*$)%\"bGF,F-F-*&F4F-F.F-!\"\"*&\"\"%F-)%\"uGF,F-F-
,$**F/F-F;F--%$sinGF1F-,&*$F*F-F-F3F7F-F77$F<,$*&F:F-,(FAF7F)F-F6F7F-F
7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "8" 0 }
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