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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 11 "Worksheet: " }{TEXT 258 12 "Lecture1.mws" }}{PARA 18 "" 0 "" 
{TEXT -1 10 "Lecture 1:" }{TEXT 260 15 " What is a PDE?" }}{PARA 19 "
" 0 "" {TEXT -1 17 "Andrew J. Bernoff" }}{PARA 18 "" 0 "" {TEXT 259 
18 "PCMI,  Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 61 "This worksheet \+
 contains the examples from the first lecture." }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 58 "Exercise 1: Verifying Some Solutions to Laplace's Eq
uation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "U:=x;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Uxx:=diff(U,x$2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "Uyy:=diff(U,y$2);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "Uxx+Uyy;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 68 "plot3d(U,x=-1..1,y=-1..1,style=patchnogrid,shading=ZHUE,axes=b
oxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "V:=x^2-y^2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Vxx:=diff(V,x$2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Vyy:=diff(V,y$2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Vxx+Vyy;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 68 "plot3d(V,x=-1..1,y=-1..1,style=patchnogrid,s
hading=ZHUE,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"W:=a*U+b*V;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Wxx:=diff(W
,x$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Wyy:=diff(W,y$2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Wxx+Wyy;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Wplot:=c*U+(1-c)*V;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "animate3d(Wplot,x=-1..1,y=-1..1,c=0
..1,style=patchnogrid,shading=ZHUE,axes=boxed,frames=50);" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 63 "Exercise 2: Verifying Solutions to the \+
Minimal Surface Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "Z:=log(sin(y)/sin(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "PDE:=(1+Zy^2)*Zxx+2*Zx*Zy*Zxy+(1+Zx^2)*Zyy;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "Zx:=diff(Z,x);Zy:=diff(Z,y);Zxx:=diff(Zx,x);
Zyy:=diff(Zy,y);Zxy:=diff(Zx,y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "PDE;simplify(PDE);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 128 "plot3d(Z,x=0..Pi,y=0..Pi,style=patchcontour,shading=
zhue,axes=boxed,grid=[100,100],title=\"Scherk's Minimal Surface\",view
=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "X:=r*cos(phi)-r
^3*cos(3*phi)/3;Y:=r*sin(phi)+r^3*sin(3*phi)/3;Z:=r^2*cos(2*phi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot3d([X,Y,Z],r=0..3,phi=-
Pi..Pi,style=patchnogrid,color=phi,axes=boxed,grid=[100,100],title=\"E
nneper's Minimal Surface\");" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 
"A Solution to the Convection Equation" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F:=x->exp(-x^2);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xi:=x-C*t;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "U:=F(xi);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "Ut:=diff(U,t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Ux:=diff(U,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "PDE:=Ut+C*Ux;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "si
mplify(PDE);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "C:=1;anima
te(U,x=-10..10,t=0..10,numpoints=200,color=blue,thickness=2,title=\"So
lution to the Transport Equation\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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