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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 34 "Surface Integrals of Vector Fiel
ds" }}{PARA 18 "" 0 "" {TEXT -1 20 "Using Maple and the " }{TEXT 256 
8 "vec_calc" }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT -1 77 "Thi
s worksheet shows how to compute surface integrals of vector fields us
ing " }{TEXT 257 5 "Maple" }{TEXT -1 9 " and the " }{TEXT 258 8 "vec_c
alc" }{TEXT -1 33 " package.  As examples we compute" }}{PARA 0 "" 0 "
" {TEXT 260 30 "* The Flux of a Magnetic Field" }{TEXT 263 0 "" }}
{PARA 0 "" 0 "" {TEXT 261 24 "* The Expansion of a Gas" }{TEXT 262 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "To st
art the " }{TEXT 259 8 "vec_calc" }{TEXT -1 41 " package, execute the \+
following commands:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar
t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "libname:=libname,\"C:
/mylib/vector\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q=C:
\\Program~Files\\Maple~6/lib6\"Q0C:/mylib/vectorF'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}{PARA 6 "" 1 "" {TEXT 
-1 33 "Package:   vec_calc   Version 4.3" }}{PARA 6 "" 1 "" {TEXT -1 
32 "For all HELP, execute: ?vec_calc" }}{PARA 6 "" 1 "" {TEXT -1 38 "T
o use aliases, execute:   vc_aliases;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 264 0 "" }{TEXT 
265 28 "The Flux of a Magnetic Field" }{TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 32 "The magnetic field of a current " }{XPPEDIT 18 0 
"i;" "6#%\"iG" }{TEXT -1 61 " moving along an infinitely long straight
 wire has magnitude " }{XPPEDIT 18 0 "B = 1/4*mu[0]*i/(Pi*r);" "6#/%\"
BG*,\"\"\"F&\"\"%!\"\"&%#muG6#\"\"!F&%\"iGF&*&%#PiGF&%\"rGF&F(" }
{TEXT -1 7 " where " }{XPPEDIT 18 0 "mu[0];" "6#&%#muG6#\"\"!" }{TEXT 
-1 19 " is a constant and " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 
155 " is the distance from the wire.  Its direction is counterclockwis
e around the wire as given by the right hand rule.  Suppose two wires \+
are parallel to the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 21 "-axi
s and located at " }{XPPEDIT 18 0 "[x, y] = [2*a, 0];" "6#/7$%\"xG%\"y
G7$*&\"\"#\"\"\"%\"aGF*\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[x, \+
y] = [-2*a, 0];" "6#/7$%\"xG%\"yG7$,$*&\"\"#\"\"\"%\"aGF+!\"\"\"\"!" }
{TEXT -1 28 " and each carries a current " }{XPPEDIT 18 0 "i;" "6#%\"i
G" }{TEXT -1 99 " but in opposite directions.  We want to find the flu
x of the magnetic field through the rectangle " }{XPPEDIT 18 0 "-b <= \+
x;" "6#1,$%\"bG!\"\"%\"xG" }{XPPEDIT 18 0 "` ` <= b;" "6#1%\"~G%\"bG" 
}{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 6 "
, and " }{XPPEDIT 18 0 "0 <= z;" "6#1\"\"!%\"zG" }{XPPEDIT 18 0 "` ` <
= c;" "6#1%\"~G%\"cG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 34 "The magnetic field of the wire at " }
{XPPEDIT 18 0 "[x, y] = [2*a, 0];" "6#/7$%\"xG%\"yG7$*&\"\"#\"\"\"%\"a
GF*\"\"!" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
74 "B1:=MF(  [x,y,z], evall(mu[0]*i/(4*Pi*((x-2*a)^2+(y)^2))*[-y,x-2*a
,0])  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G7%R6%%\"xG%\"yG%\"zG
6\"6$%)operatorG%&arrowGF+,$*&*(&%#muG6#\"\"!\"\"\"%\"iGF69%F6F6*&%#Pi
GF6,&*$),&9$F6*&\"\"#F6%\"aGF6!\"\"FAF6F6*$)F8FAF6F6F6FC#FC\"\"%F+F+F+
RF'F+F,F+,$*&*(F2F6F7F6F>F6F6*&F:F6F;F6FC#F6FGF+F+F+F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 34 "The magnetic field of the wire at " }
{XPPEDIT 18 0 "[x, y] = [-2*a, 0];" "6#/7$%\"xG%\"yG7$,$*&\"\"#\"\"\"%
\"aGF+!\"\"\"\"!" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "B2:=MF(  [x,y,z], evall(-mu[0]*i/(4*Pi*((x+2*a)^2+(y)
^2))*[-y,x+2*a,0])  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G7%R6%%
\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+,$*&*(&%#muG6#\"\"!\"\"\"%\"iG
F69%F6F6*&%#PiGF6,&*$),&9$F6*&\"\"#F6%\"aGF6F6FAF6F6*$)F8FAF6F6F6!\"\"
#F6\"\"%F+F+F+RF'F+F,F+,$*&*(F2F6F7F6F>F6F6*&F:F6F;F6FE#FEFGF+F+F+F5" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The total field is" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "B3:=MF(  [x,y,z], evall(B1(x,y,z)+B
2(x,y,z))  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B3G7%R6%%\"xG%\"yG
%\"zG6\"6$%)operatorG%&arrowGF+,&*&*(&%#muG6#\"\"!\"\"\"%\"iGF69%F6F6*
&%#PiGF6,&*$),&9$F6*&\"\"#F6%\"aGF6F6FAF6F6*$)F8FAF6F6F6!\"\"#F6\"\"%*
&#F6FGF6*&*(F2F6F7F6F8F6F6*&F:F6,&*$),&F?F6*&FAF6FBF6FEFAF6F6FCF6F6FEF
6FEF+F+F+RF'F+F,F+,&*&*(F2F6F7F6F>F6F6*&F:F6F;F6FE#FEFG*&**FFF6F2F6F7F
6FPF6F6*&F:F6FMF6FEF6F+F+F+F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 
"For the purpose of plotting the functions (and making numerical comou
tations) we make the following assignments to the variables:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=1; b:=1; c:=1; i:=1; mu[0
]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"cG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%#muG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 43 "We first plot the two functions separately:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "B1plot:=fieldplot3d(B1,-3..
3,-1..1,-1..1, grid=[9,9,2], color=blue, arrows=SLIM, orientation=[-90
,0], axes=normal, tickmarks=[7,3,3]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 133 "B2plot:=fieldplot3d(B2,-3..3,-1..1,-1..1, grid=[9,9,
2], color=red, arrows=SLIM, orientation=[-90,0], axes=normal, tickmark
s=[7,3,3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "display(arra
y([B1plot,B2plot]));" }}{PARA 13 "" 1 "" {GLPLOT3D 872 297 297 
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r" 1 2 0 1 10 0 2 1 1 1 2 1.000000 90.000000 0.000000 1 0 "Curve 1" "C
urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C
urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+
15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" 
"Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur
ve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 62 "Notice that for both fields, the arrows point in \+
the negative " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 86 "-direction \+
between the two wires.  So the fields add up.  Now we plot the total f
ield:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "B3plot:=fieldplot
3d(B3,-3..3,-1..1,-1..1, grid=[9,9,2], color=black, arrows=SLIM, orien
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[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF
[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[
qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[qF[q
F[qF[qF[q-%*AXESTICKSG6%\"\"(FjhmFjhm-%*AXESSTYLEG6#%'NORMALG-%+PROJEC
TIONG6%$!#!*F/F[qFafo" 1 2 0 1 10 0 2 1 1 4 2 1.000000 0.000000 
-90.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Befor
e we go on, we unassign the parameters:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "a:='a'; b:='b'; c:='c'; i:='i'; mu[0]:='mu[0]';" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"bGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%#muG6#\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The rectangle
 may be parametrized by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
Rect:=[u,0,v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RectG7%%\"uG\"\"!
%\"vG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 "
-b <= u;" "6#1,$%\"bG!\"\"%\"uG" }{XPPEDIT 18 0 "` ` <= b;" "6#1%\"~G%
\"bG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "0 <= v;" "6#1\"\"!%\"vG" }
{XPPEDIT 18 0 "` ` <= c;" "6#1%\"~G%\"cG" }{TEXT -1 48 ".  The tangent
 vectors and the normal vector are" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "Ru:=diff(Rect,u); Rv:=diff(Rect,v); N:=Ru &x Rv;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RuG7%\"\"\"\"\"!F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#RvG7%\"\"!F&\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG7%\"\"!!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "Notice that the normal points in the negative " }{XPPEDIT 18 0 
"y;" "6#%\"yG" }{TEXT -1 85 "-direction, which is the direction we saw
 in the plot that the magnetic field points." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "On the rectangle, the magnetic \+
field is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "B3(u,0,v); simp
lify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!,&*&*&&%#muG6#F$\"
\"\"%\"iGF+F+*&%#PiGF+,&%\"uGF+*&\"\"#F+%\"aGF+F+F+!\"\"#F4\"\"%*&*(#F
+F6F+F(F+F,F+F+*&F.F+,&F0F+*&F2F+F3F+F4F+F4F+F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%\"\"!,$*&*(&%#muG6#F$\"\"\"%\"iGF+%\"aGF+F+*(%#PiGF+,
&%\"uGF+*&\"\"#F+F-F+F+F+,&F1!\"\"*&F3F+F-F+F+F+F5F5F$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 14 "So the flux is" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 48 "interface(showassumed=0); assume(0<b,b<2*a,0<c);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Muint(B3(u,0,v) &. N,u=-b
..b,v=0..c); Flux:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int
G6$-F$6$*&*(&%#muG6#\"\"!\"\"\"%\"iGF.%#a|irGF.F.*(%#PiGF.,&%\"uGF.*&
\"\"#F.F0F.F.F.,&F4!\"\"*&F6F.F0F.F.F.F8/F4;,$%#b|irGF8F=/%\"vG;F-%#c|
irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FluxG,$*&**&%#muG6#\"\"!\"\"
\"%\"iGF,,&-%#lnG6#,&%#b|irGF,*&\"\"#F,%#a|irGF,F,F,-F06#,&F3!\"\"*&F5
F,F6F,F,F:F,%#c|irGF,F,%#PiGF:#F,F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "Flux:=combine(%,ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%FluxG*&**&%#muG6#\"\"!\"\"\"%\"iGF+%#c|irGF+-%#lnG6#*$-%%sqrtG6#
*&,&%#b|irGF+*&\"\"#F+%#a|irGF+F+F+,&F7!\"\"*&F9F+F:F+F+F<F+F+F+%#PiGF
<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Note:  There is no guarante
e that if you re-execute these commands that Maple will give the answe
r in the same form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 59 "Another way to compute this surface integral is to use \+
the " }{TEXT 267 18 "Surface_int_vector" }{TEXT -1 23 " command (or it
s alias " }{TEXT 268 3 "Siv" }{TEXT -1 11 ") from the " }{TEXT 269 8 "
vec_calc" }{TEXT -1 81 " package which works directly with the paramet
rized surface and the vector field:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "Ruv:=MF([u,v],Rect);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$RuvG7%R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF*9$F*F*F*\"\"!RF'F*F
+F*9%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Siv(B3,Ruv,u
=-b..b,v=0..c); Flux:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$
IntG6$-F$6$*&*(&%#muG6#\"\"!\"\"\"%\"iGF.%#a|irGF.F.*(%#PiGF.,&%\"uGF.
*&\"\"#F.F0F.F.F.,&F4!\"\"*&F6F.F0F.F.F.F8/F4;,$%#b|irGF8F=/%\"vG;F-%#
c|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FluxG,$*&**&%#muG6#\"\"!\"
\"\"%\"iGF,,&-%#lnG6#,&%#b|irGF,*&\"\"#F,%#a|irGF,F,F,-F06#,&F3!\"\"*&
F5F,F6F,F,F:F,%#c|irGF,F,%#PiGF:#F,F5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 108 "There is a second way to compute this flux.  The magneti
c field is solenoidal because it is divergence-free:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "DIV(B3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "So the magnetic field \+
has a vector potential, " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 3 ".
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "VEC_POT(B3,'A');" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "A(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&**&
%#muG6#\"\"!\"\"\"%\"iGF+,&%\"xGF+*&\"\"#F+%#a|irGF+F+F+%\"zGF+F+*&%#P
iGF+,&*$)F-F0F+F+*$)%\"yGF0F+F+F+!\"\"#F;\"\"%*&*,#F+F=F+F'F+F,F+,&F.F
+*&F0F+F1F+F;F+F2F+F+*&F4F+,&*$)FAF0F+F+F8F+F+F;F+,&*&**F'F+F,F+F:F+F2
F+F+*&F4F+F5F+F;F<*&*,F@F+F'F+F,F+F:F+F2F+F+*&F4F+FDF+F;F+F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "By Stokes' Theorem, the flux is t
he line integral of the vector potential around the boundary curve.  H
ere the boundary consists of four line segments:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "r1:=MF([u],[u,0,0]); #for u=-b..b" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#r1G7%R6#%\"uG6\"6$%)operatorG%&arrowGF)9$
F)F)F)\"\"!F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "r2:=MF([v]
,[b,0,v]); #for v=0..c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G7%R6#%
\"vG6\"6$%)operatorG%&arrowGF)%#b|irGF)F)F)\"\"!RF'F)F*F)9$F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r3:=MF([u],[-u,0,c]); #for u
=-b..b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3G7%R6#%\"uG6\"6$%)opera
torG%&arrowGF),$9$!\"\"F)F)F)\"\"!RF'F)F*F)%#c|irGF)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r4:=MF([v],[-b,0,c-v]); #for v=0..c
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r4G7%R6#%\"vG6\"6$%)operatorG%&
arrowGF),$%#b|irG!\"\"F)F)F)\"\"!RF'F)F*F),&%#c|irG\"\"\"9$F/F)F)F)" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "So the flux is the total line in
tegral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Liv(A,r1,u=-b..b
)+Liv(A,r2,v=0..c)+Liv(A,r3,u=-b..b)+Liv(A,r4,v=0..c); Flux:=value(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$IntG6$\"\"!/%\"uG;,$%#b|irG!
\"\"F,\"\"\"*&\"\"#F.-F%6$F'/%\"vG;F'%#c|irGF.F.-F%6$*&**&%#muG6#F'F.%
\"iGF.F6F.%#a|irGF.F.*(%#PiGF.,&F)F.*&F0F.F?F.F.F.,&F)F-*&F0F.F?F.F.F.
F-F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FluxG,$*&**&%#muG6#\"\"!
\"\"\"%\"iGF,,&-%#lnG6#,&%#b|irGF,*&\"\"#F,%#a|irGF,F,F,-F06#,&F3!\"\"
*&F5F,F6F,F,F:F,%#c|irGF,F,%#PiGF:#F,F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 22 "The Expans
ion of a Gas" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The velocity field
 of a gas is " }{XPPEDIT 18 0 "V = [x*y^2, y*z^2, z*x^2];" "6#/%\"VG7%
*&%\"xG\"\"\"*$%\"yG\"\"#F(*&F*F(*$%\"zGF+F(*&F.F(*$F'F+F(" }{TEXT -1 
70 " .  We want to find the expansion of the fluid out through the sph
ere " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 4;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"y
GF'F(*$%\"zGF'F(\"\"%" }{TEXT -1 21 " which is defined as " }{XPPEDIT 
18 0 "Expansion = Int(`V.`,S);" "6#/%*ExpansionG-%$IntG6$%#V.G%\"SG" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = Int(Int(`V.N`,theta = 0 .. 2*Pi),
phi = 0 .. Pi);" "6#/%\"~G-%$IntG6$-F&6$%$V.NG/%&thetaG;\"\"!*&\"\"#\"
\"\"%#PiGF1/%$phiG;F.F2" }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We enter the fluid velocity as \+
a function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "V:=MF([x,y,z
],[x*y^2,y*z^2,z*x^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG7%R6%
%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+*&9$\"\"\")9%\"\"#F1F+F+F+RF'
F+F,F+*&F3F1)9&F4F1F+F+F+RF'F+F,F+*&F8F1)F0F4F1F+F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "The sphere may be parametrized as" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Rsph:=[2*sin(phi)*cos(theta)
,2*sin(phi)*sin(theta),2*cos(phi)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%RsphG7%,$*&-%$sinG6#%$phiG\"\"\"-%$cosG6#%&thetaGF,\"\"#,$*&F(F,-F
)F/F,F1,$-F.F*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "So the tangen
t vectors are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R[theta]:
=diff(Rsph,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%&theta
G7%,$*&-%$sinG6#%$phiG\"\"\"-F,F&F/!\"#,$*&F+F/-%$cosGF&F/\"\"#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R[phi]:=diff(Rsph,phi);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%$phiG7%,$*&-%$cosGF&\"\"\"
-F,6#%&thetaGF-\"\"#,$*&F+F--%$sinGF/F-F1,$-F5F&!\"#" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 24 "and the normal vector is" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "N:= R[theta] &x R[phi];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG7%,$*&)-%$sinG6#%$phiG\"\"#\"\"\"-%$cosG6#%&theta
GF.!\"%,$*&F(F.-F*F1F.F3,&*(F)F.)F6F-F.-F0F+F.F3**\"\"%F.F)F.)F/F-F.F:
F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We simplify the third \+
component:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "N[3]:=simplif
y(N[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"NG6#\"\"$,$*&-%$sinG6
#%$phiG\"\"\"-%$cosGF,F.!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "T
he components of " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 39 " are ne
gative in the first octant.  So " }{XPPEDIT 18 0 "N;" "6#%\"NG" }
{TEXT -1 58 " points inward.  We want the flux outward.  So we reverse
 " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "N:=-N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"N
G7%,$*&)-%$sinG6#%$phiG\"\"#\"\"\"-%$cosG6#%&thetaGF.\"\"%,$*&F(F.-F*F
1F.F3,$*&F)F.-F0F+F.F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "On the \+
sphere, the velocity is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
VR:=V(op(Rsph));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VRG7%,$*()-%$si
nG6#%$phiG\"\"$\"\"\"-%$cosG6#%&thetaGF.)-F*F1\"\"#F.\"\"),$*(F)F.F4F.
)-F0F+F5F.F6,$*(F:F.)F)F5F.)F/F5F.F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 63 "Note:  We need the op command to strip the square brackets off \+
" }{XPPEDIT 18 0 "Rsph;" "6#%%RsphG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The dot product of the
 velocity field and the normal is:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "VN:=VR &. N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VN
G,$*(,&!\"\"\"\"\"*$)-%$cosG6#%$phiG\"\"#F)F)F)-%$sinGF.F),,*$)-F-6#%&
thetaGF0F)F)*$)F6\"\"%F)F(*&F+F)F5F)F(*&F:F)F+F)F)F*F)F)!#K" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "So the expansion is:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Muint(VN,theta=0..2*Pi,phi=0..Pi); \+
Expansion:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$
,$*(,&!\"\"\"\"\"*$)-%$cosG6#%$phiG\"\"#F,F,F,-%$sinGF1F,,,*$)-F06#%&t
hetaGF3F,F,*$)F9\"\"%F,F+*&F.F,F8F,F+*&F=F,F.F,F,F-F,F,!#K/F;;\"\"!,$%
#PiGF3/F2;FDFF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ExpansionG,$%#PiG
#\"$G\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Another way to co
mpute this surface integral is to use the " }{TEXT 270 18 "Surface_int
_vector" }{TEXT -1 23 " command (or its alias " }{TEXT 271 3 "Siv" }
{TEXT -1 11 ") from the " }{TEXT 272 8 "vec_calc" }{TEXT -1 81 " packa
ge which works directly with the parametrized surface and the vector f
ield:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R:=MF([theta,phi],
Rsph);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%R6$%&thetaG%$phiG6\"
6$%)operatorG%&arrowGF*,$*&-%$sinG6#9%\"\"\"-%$cosG6#9$F4\"\"#F*F*F*RF
'F*F+F*,$*&F0F4-F1F7F4F9F*F*F*RF'F*F+F*,$-F6F2F9F*F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Siv(V,R,theta=0..2*Pi,phi=0..Pi); E
xpansion:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,
$*(,&!\"\"\"\"\"*$)-%$cosG6#%$phiG\"\"#F,F,F,-%$sinGF1F,,,*$)-F06#%&th
etaGF3F,F,*$)F9\"\"%F,F+*&F.F,F8F,F+*&F=F,F.F,F,F-F,F,\"#K/F;;\"\"!,$%
#PiGF3/F2;FDFF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ExpansionG,$%#PiG
#!$G\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Notice there is a \+
minus sign difference in the answer because we did not reverse the nor
mal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "
There is a second way to compute this expansion.  By Gauss' theorem, t
he expansion is the volume integral of the divergence of the velocity \+
over the interior of the sphere.  In this case, the divergence is:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "DIV(V);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF(,(*$)9%\"\"
#\"\"\"F1*$)9&F0F1F1*$)9$F0F1F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 " which in spherical coordinates is" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "divV:=rho^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%divVG*$)%$rhoG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 
"The spherical volume Jacobian is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "J:=rho^2*sin(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"JG*&)%$rhoG\"\"#\"\"\"-%$sinG6#%$phiGF)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 19 "So the expansion is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "Muint(divV*J,rho=0..2,theta=0..2*Pi,phi=0..Pi); Expan
sion:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$
*&)%$rhoG\"\"%\"\"\"-%$sinG6#%$phiGF./F,;\"\"!\"\"#/%&thetaG;F5,$%#PiG
F6/F2;F5F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ExpansionG,$%#PiG#\"$
G\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "11
 12 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 
1 }