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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "PDE's and Change of Varia
bles" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "First order example from \+
class." }}{PARA 0 "" 0 "" {TEXT -1 48 "This verifies that a function f
(x,y) =g(x^2+y^2)" }}{PARA 0 "" 0 "" {TEXT -1 33 "solves the PDE y f_x
 - x f_y = 0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "expression := y*di
ff(f(x,y),x) - x*diff(f(x,y),y);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "subs(f(x,y) = g(x^2+y^2),expression
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+expressionG,&*&%\"yG\"\"\"-%%
diffG6$-%\"fG6$%\"xGF'F/F(F(*&F/F(-F*6$F,F'F(!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*&%\"yG\"\"\"-%%diffG6$-%\"gG6#,&*$)%\"xG\"\"#F&F&*
$)F%F1F&F&F0F&F&*&F0F&-F(6$F*F%F&!\"\"" }}}{EXCHG {PARA 18 "" 0 "" 
{TEXT 256 17 "The wave equation" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
74 "We want to look at the change of variables u =x+y, v=x-y in this e
quation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "waveqn := diff(f(x,y),x
,x) - diff(f(x,y),y,y) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'wave
qnG/,&-%%diffG6$-%\"fG6$%\"xG%\"yG-%\"$G6$F-\"\"#\"\"\"-F(6$F*-F06$F.F
2!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This means that f(
x,y) = g(u,v) = g(x+y,x-y)." }}{PARA 0 "" 0 "" {TEXT -1 63 "Without th
e simplify, this doesn't really give us what we want." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "waveqn2 := subs(f(x,y)=g(x+y,x-y),waveqn);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(waveqn2G/,&-%%diffG6$-%\"gG6$,&%\"x
G\"\"\"%\"yGF/,&F.F/F0!\"\"-%\"$G6$F.\"\"#F/-F(6$F*-F46$F0F6F2\"\"!" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "We see that the equation is of m
uch simpler form now." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "waveqn2 :=
 simplify(subs(f(x,y)=g(x+y,x-y),waveqn));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(waveqn2G/,$--&%\"DG6$\"\"\"\"\"#6#%\"gG6$,&%\"xGF,%
\"yGF,,&F2F,F3!\"\"\"\"%\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "
Now we express the new equation entirely in terms of u and v." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "invtrans := solve(\{u=x+y,v=x-y\},
\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)invtransG<$/%\"xG,&%\"
vG#\"\"\"\"\"#*&F*F+%\"uGF+F+/%\"yG,&F.F**&#F+F,F+F)F+!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The new equation is g_uv = 0." }}
{PARA 0 "" 0 "" {TEXT -1 75 "(In general we might need more sophistica
ted simplification at this stage.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "waveqn3 := subs(invtrans,waveqn2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(waveqn3G/,$--&%\"DG6$\"\"\"\"\"#6#%\"gG6$%\"uG%\"vG\"\"%\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 256 "" 0 "" {TEXT 257 39 "Laplace's equation in Polar C
oordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 48 "Laplace's equation in two variables is given by " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "lapeqn := diff(f(x,y),x,x) + diff(f(x,y),y,y)=
0;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 0 "" {TEXT -1 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lapeqnG/,&-%%diffG6$-%\"fG6$%\"xG
%\"yG-%\"$G6$F-\"\"#\"\"\"-F(6$F*-F06$F.F2F3\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 95 "It is useful in applications to rewrite this equat
ion in polar coordinates.  To this end we set" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "lapeqn2 := \+
simplify(subs(f(x,y) = g(sqrt(x^2 + y^2),arctan(y/x)),lapeqn));" }}
{PARA 0 "" 0 "" {TEXT -1 87 "This doesn't look very simplified!  Let's
 try coverting x and y into polar coordinates:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(lapeqn2G/*&,,*()%\"
xG\"\"#\"\"\"-%%sqrtG6#,&*$F)F,F,*$)%\"yGF+F,F,F,--&%\"DG6$F,F,6#%\"gG
6$*$F-F,-%'arctanG6#*&F4F,F*!\"\"F,F,*&--&F86#F,F:F<F,F)F,F,*&F-F,--&F
86$F+F+F:F<F,F,*(F-F,F5F,F3F,F,*&FDF,F3F,F,F,*$)F0#\"\"$F+F,FB\"\"!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "lapeqn3 := simplify(subs(x
=r*cos(theta),y=r*sin(theta),lapeqn2));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%(lapeqn3G/*&*&,(*&-%%csgnG6#%\"rG\"\"\"--&%\"DG6$\"\"#F46#%\"g
G6$*&F*F.F-F.-%'arctanG6#*&-%$sinG6#%&thetaGF.-%$cosGF?!\"\"F.F.*()F-F
4F.F*F.--&F26$F.F.F5F7F.F.*&--&F26#F.F5F7F.F-F.F.F.F*F.F.*$FEF.FC\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Ok, we're getting somewhere.
  However, there's still that annoying \"csgn(r)\" that arises because
 the square root of r*r is " }}{PARA 0 "" 0 "" {TEXT -1 117 "|r| rathe
r than just r.   We know that r can be taken positive in polar coordin
ates, but Maple doesn't.  Furthermore," }}{PARA 0 "" 0 "" {TEXT -1 21 
"we see easily that   " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 
0 "" 0 "" {TEXT -1 66 "      arctan(sin(theta)/cos(theta))  =  arctan(
tan(theta) = theta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 113 "Maple misses that point--or seems to.  Actually, Maple
 is probably only acting out of undue caution.  In general," }}{PARA 
0 "" 0 "" {TEXT -1 121 "arctan(tan(theta)) might be equal to \"theta +
 k*pi\" rather than just theta.  In practice, we regard this sort of t
hing as" }}{PARA 0 "" 0 "" {TEXT -1 155 "a technical point and ignore \+
it until it gets us into trouble.  Maple tends to avoid trouble at all
 costs.  Rather than force Maple to see things our way, " }}{PARA 0 "
" 0 "" {TEXT -1 41 "we can just rewrite the equation by hand:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "la
peqn4 := r^2 * diff(g(r,theta),r,r) + \n           r * diff(g(r,theta)
,r) +\n           r^(-2) * diff(g(r,theta),theta,theta) =            0
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lapeqn4G/,(*&)%\"rG\"\"#\"\"\"
-%%diffG6$-%\"gG6$F)%&thetaG-%\"$G6$F)F*F+F+*&F)F+-F-6$F/F)F+F+*&-F-6$
F/-F46$F2F*F+*$F(F+!\"\"F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
146 "(Maple does have an assume facility that lets you announce things
 like \"assume(r>0)\" to get rid of the csgn, but there are real syste
m limitations" }}{PARA 0 "" 0 "" {TEXT -1 25 "on how far this gets you
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "In general, harmonic functi
ons (i.e. solutions of Laplace's equation) depend on r and theta, but \+
it is interesting to see" }}{PARA 0 "" 0 "" {TEXT -1 61 "if there are \+
any harmonic functions h that depend on r alone." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rlapeqn := simplify
(subs(g(r,theta) = h(r),lapeqn4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%(rlapeqnG/,&*&)%\"rG\"\"#\"\"\"-%%diffG6$-%\"hG6#F)-%\"$G6$F)F*F+F+*
&F)F+-F-6$F/F)F+F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This i
s an ordinary differential equation with a simple solution." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(rl
apeqn,h(r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"rG,&%$_C1G
\"\"\"*&%$_C2GF*-%#lnGF&F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "
The punchline is that any radially symmetric harmonic function is an a
ffine function (i.e. linear function plus constant)" }}{PARA 0 "" 0 "
" {TEXT -1 9 "of log r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 51 "Exercise:  Laplace's equation in three variables is
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 
"laplace := diff(f(x,y,z),x,x) + diff(f(x,y,z),y,y)\n           + diff
(f(x,y,z),z,z) = 0;    " }}{PARA 0 "" 0 "" {TEXT -1 121 "Try turning t
his into an equation in spherical coordinates!  What harmonic function
s are functions of the radial variable" }}{PARA 0 "" 0 "" {TEXT -1 11 
"\"rho\" only?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(laplaceG/,(-%%dif
fG6$-%\"fG6%%\"xG%\"yG%\"zG-%\"$G6$F-\"\"#\"\"\"-F(6$F*-F16$F.F3F4-F(6
$F*-F16$F/F3F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }