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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Unit 4: Vector Calculus" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Chapter \+
20: Additional Vector Differential Operators" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Section 20.1:  divergence and i
ts meaning" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ
ts reserved.  No part of this publication may be reproduced, stored in
 a retrieval system, or transmitted, in any form or by any means, elec
tronic, mechanical, photocopying, recording, or otherwise, without the
 prior written permission of the publisher. Printed in the United Stat
es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "with(linalg):\nwith(plots):\nwith(student):
\nwith(plottools):\nread(`pvac.txt`):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Definition \+
20.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "Given the vector field" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }
{TEXT 257 4 "F = " }{XPPEDIT 18 0 "u(x,y,z)" "6#-%\"uG6%%\"xG%\"yG%\"z
G" }{TEXT -1 1 " " }{TEXT 292 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "
v(x,y,z)" "6#-%\"vG6%%\"xG%\"yG%\"zG" }{TEXT -1 1 " " }{TEXT 293 1 "j
" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "w(x,y,z)" "6#-%\"wG6%%\"xG%\"yG%\"
zG" }{TEXT -1 1 " " }{TEXT 294 1 "k" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "the " }{TEXT 296 10 "div
ergence" }{TEXT -1 4 " of " }{TEXT 295 1 "F" }{TEXT -1 15 "  at the po
int " }{XPPEDIT 18 0 "``(x,y,z)" "6#-%!G6%%\"xG%\"yG%\"zG" }{TEXT -1 
14 " is the scalar" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "
" {TEXT 299 1 "f" }{TEXT -1 1 " " }{TEXT 300 1 "." }{TEXT -1 1 " " }
{TEXT 297 1 "F" }{TEXT -1 7 " = div(" }{TEXT 298 1 "F" }{TEXT -1 4 ") \+
= " }{XPPEDIT 18 0 "u[x](x,y,z\}+v[y](x,y,z)+w[z](x,y,z)" "6#,(<#-&%\"
uG6#%\"xG6%F)%\"yG%\"zG\"\"\"-&%\"vG6#F+6%F)F+F,F--&%\"wG6#F,6%F)F+F,F
-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 28 "The divergence of the field " }{TEXT 301 1 "F" }{TEXT -1 
236 " is a scalar which measures the \"spread\" of the field at each p
oint.  Notational clarifications, as well as xxamples and calculations
 designed to illuminate this claim about divergence measuring \"spread
\" of the field at a point, follow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 49 "Maple computes the divergence of the vec
tor field" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "F := vector([u(x,y,z), v(x,y,z), w(x,y,z)]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "with the command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diverge(F, [x,y,z]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 12 "Example 20.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "The field " }{TEXT 256 1 "F" }{TEXT -1 9 
" given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "F := vector([x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "graphed in Fi
gure 20.1 (below), appears to be \"flowing\" radially outward, with fl
ow lines " }{XPPEDIT 18 0 "y=alpha*x" "6#/%\"yG*&%&alphaG\"\"\"%\"xGF'
" }{TEXT -1 44 ", the solution of the differential equations" }}{PARA 
266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`x'`(p)=x" "6#/-%#x'G6#%\"p
G%\"xG" }{TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`y'`(p)=y" "6#/-%#y'G6#%\"pG%\"yG" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 12 "contained in" }}{PARA 268 "" 0 "" {TEXT -1 1 " " }
{TEXT 302 1 "R" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "``(p)" "6#-%!G6#%\"pG
" }{TEXT -1 3 " = " }{TEXT 303 1 "F" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Figure 20.1 is generate
d in Maple by executing the following commands." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 607 "p1 := circ
le([0,0],1/2,color=red):\np2 := circle([0,0],3/2,color=green):\nf1 := \+
z -> arrow([.5*cos(z),.5*sin(z)],vector([.5*cos(z),.5*sin(z)]), .05,.2
,.2, color=red):\nf2 := z -> arrow([1.5*cos(z),1.5*sin(z)],vector([1.5
*cos(z),1.5*sin(z)]), .05,.2,.2, color=green):\np3:=plot([[t,t,t=-2..2
],[-t,t,t=-2..2],[t,t*tan(Pi/12),t=-2.5..2.5], [t,t*tan(5*Pi/12),t=-.7
.. .7],[t,t*tan(7*Pi/12),t=-.7.. .7],[t,t*tan(11*Pi/12),t=-2.5..2.5]],
 color=black):\ndisplay([p||(1..3),seq(f1(Pi/6*k),k=0..11),seq(f2(Pi/6
*k),k=0..11)], scaling=constrained, xtickmarks=7, ytickmarks=7, labels
=[`   x`,y], labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The paramet
er " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 33 " is eliminated from th
e equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dp=x(p)" "6#/*&%#dxG\"\"\"%#dpG!\"\"
-%\"xG6#%\"pG" }{TEXT -1 1 " " }}{PARA 270 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "dy/dp=y(p)" "6#/*&%#dyG\"\"\"%#dpG!\"\"-%\"yG6#%\"pG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "by writing" }}{PARA 271 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(dy/dp)/``(dx/dp)=dy/dx" "6#/
*&-%!G6#*&%#dyG\"\"\"%#dpG!\"\"F*-F&6#*&%#dxGF*F+F,F,*&F)F*F0F," }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "`y'`/`x'`=y/x" "6#/*&%#y'G\"\"\"%#x'G
!\"\"*&%\"yGF&%\"xGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "
and the solution " }{XPPEDIT 18 0 "y=alpha*x" "6#/%\"yG*&%&alphaG\"\"
\"%\"xGF'" }{TEXT -1 14 " is immediate." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 42 "The field arises from the scalar pot
ential" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x,y)=1/2
" "6#/-%\"uG6$%\"xG%\"yG*&\"\"\"F*\"\"#!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "``(x^2+y^2)" "6#-%!G6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "where the minus sign ha
s been ignored." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "The level curves of " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$
%\"xG%\"yG" }{TEXT -1 66 " are concentric circles orthogonal to the fl
ow lines of the field " }{TEXT 304 1 "F" }{TEXT -1 25 ", as seen in Fi
gure 20.1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 22 "The divergence is div(" }{TEXT 305 1 "F" }{TEXT -1 164 ") = 2. \+
 The constant and positive value of the divergence signals the uniform
 spread of the field, evidenced by the flow pointing radially outwards
 from the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 53 "To implement these calculations in Maple, begin with " }
{TEXT 268 1 "R" }{TEXT -1 42 ", the radius-vector form of the flow lin
e," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "R := vector([x(p), y(p)]);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and differ
entiate with respect to the parameter " }{TEXT 269 1 "t" }{TEXT -1 28 
" to form the tangent vector." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`R'` := map(diff,R,p);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Equating the vectors " }{TEXT 273 1 "R" }{TEXT -1 6 "' an
d " }{TEXT 274 1 "F" }{TEXT -1 60 " means equating their components, a
ccomplished in Maple with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "q := equate(`R'`,F);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 "This set of differential equations is then solved via" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q
1 := dsolve(q, \{x(p),y(p)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Forming the ratio " }
{XPPEDIT 18 0 "y/x=C[2]/C[1]" "6#/*&%\"yG\"\"\"%\"xG!\"\"*&&%\"CG6#\"
\"#F&&F+6#F&F(" }{TEXT -1 26 " eliminates the parameter " }{XPPEDIT 
18 0 "p" "6#%\"pG" }{TEXT -1 44 " and shows analytically that the flow
 lines " }{XPPEDIT 18 0 "y=alpha*x" "6#/%\"yG*&%&alphaG\"\"\"%\"xGF'" 
}{TEXT -1 36 " are rays emanating from the origin." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Maple claims " }{TEXT 
275 1 "F" }{TEXT -1 24 " has a scalar potential " }{XPPEDIT 18 0 "u(x,
y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 37 " since the potential command \+
returns " }{TEXT 276 4 "true" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "potential(F, [x,y
], 'u');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{TEXT 277 1 "F" }{TEXT -1 44 " is
 conservative with the potential function" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "u;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "w
here the physicist's minus sign has been ignored.  Thus, " }{TEXT 278 
1 "F" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grad(u, [x,y]);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The
 level curves of " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }
{TEXT -1 47 " are orthogonal to the flow lines of the field " }{TEXT 
258 1 "F" }{TEXT -1 40 "., as suggested by the following figure." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 118 "p4 := contourplot(u, x = -1..1, y = -1..1, contours=[.1, .3, .5
], color=black, scaling=constrained):\ndisplay([p3,p4]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 74 "Finally, the divergence of this now throroughly-familiar field \+
is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "diverge(F, [x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "Divergence \+
as Limiting Value of Flux per Unit Area" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "We next give a physical inte
rpretation of the divergence based on the notion of flux through a clo
sed curve " }{TEXT 306 1 "C" }{TEXT -1 50 ".  It demonstrates that in \+
the limit as the curve " }{TEXT 307 1 "C" }{TEXT -1 20 " shrinks to a \+
point " }{TEXT 308 1 "P" }{TEXT -1 37 ", the ratio of the flux of the \+
field " }{TEXT 259 1 "F" }{TEXT -1 9 " through " }{TEXT 309 1 "C" }
{TEXT -1 33 " divided by the area enclosed in " }{TEXT 310 1 "C" }
{TEXT -1 38 ", becomes the divergence of the field " }{TEXT 260 1 "F" 
}{TEXT -1 14 " evaluated at " }{TEXT 311 1 "P" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "To comput
e the flux of the arbitrary plane vector " }{TEXT 261 1 "F" }{TEXT -1 
3 " = " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 " \+
" }{TEXT 312 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG
6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 313 1 "j" }{TEXT -1 10 ", that is
," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "F := vector([f(x,y),g(x,y)]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "through
 " }{TEXT 314 1 "C" }{TEXT -1 21 ", a circle of radius " }{TEXT 279 1 
"a" }{TEXT -1 24 ", centered at the point " }{TEXT 315 1 "P" }{TEXT 
-1 24 " whose coordinates are (" }{XPPEDIT 18 0 "x[0],y[0]" "6$&%\"xG6
#\"\"!&%\"yG6#F&" }{TEXT -1 56 "), write the parametric representation
 of this circle as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "X := x[0]+a*cos(p);\nY := y[0]+a*sin(p);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 11 "so that on " }{TEXT 316 1 "C" }{TEXT -1 9 " we have " }
{TEXT 317 1 "F" }{TEXT -1 3 " = " }{TEXT 318 1 "F" }{XPPEDIT 18 0 "``(
x[0]+a*cos(p),y[0]+a*sin(p))" "6#-%!G6$,&&%\"xG6#\"\"!\"\"\"*&%\"aGF+-
%$cosG6#%\"pGF+F+,&&%\"yG6#F*F+*&F-F+-%$sinG6#F1F+F+" }{TEXT -1 10 ", \+
that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "Fc := subs(x=X,y=Y, op(F));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "As the c
ircle " }{TEXT 319 1 "C" }{TEXT -1 22 " shrinks to the point " }{TEXT 
320 1 "P" }{TEXT -1 13 ", the radius " }{TEXT 262 1 "a" }{TEXT -1 30 "
 goes to zero.  Hence, expand " }{TEXT 263 1 "F" }{TEXT -1 26 " in a T
aylor series about " }{XPPEDIT 18 0 "a=0" "6#/%\"aG\"\"!" }{TEXT -1 
11 ", obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" 
{TEXT -1 1 " " }{TEXT 321 1 "F" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "MATR
IX([[f(x[0],y[0])+(f[x](x[0],y[0])(cos(p)+f[y](x[0],y[0])*sin(p))*a+O(
a*2)],[g(x[0],y[0])+(g[x](x[0],y[0])(cos(p)+f[y](x[0],y[0])*sin(p))*a+
O(a*2)]])" "6#-%'MATRIXG6#7$7#,&-%\"fG6$&%\"xG6#\"\"!&%\"yG6#F/\"\"\",
&*&--&F*6#F-6$&F-6#F/&F16#F/6#,&-%$cosG6#%\"pGF3*&-&F*6#F16$&F-6#F/&F1
6#F/F3-%$sinG6#FDF3F3F3%\"aGF3F3-%\"OG6#*&FQF3\"\"#F3F3F37#,&-%\"gG6$&
F-6#F/&F16#F/F3,&*&--&FZ6#F-6$&F-6#F/&F16#F/6#,&-FB6#FDF3*&-&F*6#F16$&
F-6#F/&F16#F/F3-FO6#FDF3F3F3FQF3F3-FS6#*&FQF3FVF3F3F3" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "map(taylor,Fc,a=0,2)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 107 "Rather than try to use Maple to simplify the notati
on, just enter the expanded, and simplified versions of " }{TEXT 270 
1 "f" }{TEXT -1 5 " and " }{TEXT 271 1 "g" }{TEXT -1 9 ", writing" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 79 "fC := f[0]+(f[x]*cos(p)+f[y]*sin(p))*a;\ngC := g[0]+(g[x]*cos(p)
+g[y]*sin(p))*a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The integrand for the flux integra
l for " }{TEXT 264 1 "F" }{TEXT -1 13 " is given by " }{TEXT 265 1 "F
" }{TEXT 266 1 "." }{TEXT 267 1 "N" }{TEXT -1 1 " " }{TEXT 272 2 "ds" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "f*dy-g*dx" "6#,&*&%\"fG\"\"\"%#dyGF&
F&*&%\"gGF&%#dxGF&!\"\"" }{TEXT -1 20 ", formed in Maple by" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "`
fdy - gdx` := fC*diff(Y,p) - gC*diff(X,p);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The flux in
tegral " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "q := Int(`fdy - gdx`, p=0..2*Pi);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "is \+
then divided by " }{XPPEDIT 18 0 "Pi*a^2" "6#*&%#PiG\"\"\"*$%\"aG\"\"#
F%" }{TEXT -1 34 ", the area enclosed by the circle " }{TEXT 322 1 "C
" }{TEXT -1 23 ", and evaluated, giving" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(value(q/(Pi*a
^2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 30 "As predicted, in the limit as " }{XPPEDIT 18 0 
"a" "6#%\"aG" }{TEXT -1 58 " goes to zero, the limiting ratio of flux \+
to area becomes " }{XPPEDIT 18 0 "f[x]+g[y]" "6#,&&%\"fG6#%\"xG\"\"\"&
%\"gG6#%\"yGF(" }{TEXT -1 20 ", the divergence of " }{TEXT 323 1 "F" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Soleno
idal Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 144 "Because the magnetic field set up by a solenoid (coil of
 wire) has zero divergence in the space surrounded by the coil, physic
ists use the term " }{TEXT 280 10 "solenoidal" }{TEXT -1 72 " for fiel
ds whose divergence is everywhere zero.  For example, the field" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "F := vector([x,-y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "is solenoidal since its di
vergence vanishes, as the following shows." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diverge(F,[x,y]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 28 "On the other hand, the field" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F := vector
([x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "is not solenoidal, since div(" }{TEXT 
281 1 "F" }{TEXT -1 18 ") = 2, as shown by" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diverge(F,[x,y]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 91 "Connections between conservative fields and solenoid
al fields are detailed in Section 21.5." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 325 1 "F" }{TEXT -1 190 "
 is the field of velocity vectors for a steady fluid flow with constan
t density, then its divergence must be zero, as we discuss in Section \+
23.3.  But a steady flow with constant density is " }{TEXT 324 14 "inc
ompressible" }{TEXT -1 67 ", so the analog of the solenoidal field is \+
the incompressible flow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 4 "The " }{TEXT 326 9 "laplacian" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 327 9 "laplacian" }
{TEXT -1 15 " of the scalar " }{XPPEDIT 18 0 "u(x,y,z)" "6#-%\"uG6%%\"
xG%\"yG%\"zG" }{TEXT -1 20 " is the scalar field" }}{PARA 274 "" 0 "" 
{XPPEDIT 18 0 "div(grad(u))" "6#-%$divG6#-%%gradG6#%\"uG" }{TEXT -1 5 
"  =  " }{TEXT 331 1 "f" }{TEXT -1 1 " " }{TEXT 328 1 "." }{TEXT -1 2 
" (" }{TEXT 332 1 "f" }{TEXT -1 1 " " }{TEXT 329 1 "u" }{TEXT -1 4 ") \+
= " }{TEXT 333 1 "f" }{XPPEDIT 18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 1 
" " }{TEXT 330 1 "u" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "th
e divergence of the gradient field of " }{TEXT 334 1 "u" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 28 "The laplacian of the scalar " }{XPPEDIT 
18 0 "u(x,y,z)" "6#-%\"uG6%%\"xG%\"yG%\"zG" }{TEXT -1 21 " is given in
 Maple by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "u := 'u':\nlaplacian(u(x,y,z),[x,y,z]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 70 "To obtain this same result at the divergence of the gradient, c
ompute " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "diverge(grad(u(x,y,z),[x,y,z]),[x,y,z]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 87 "In cartesian coordinates, the del operator (Section 18.2) is (u
sing an equation editor)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 
"" 0 "" {TEXT -1 1 " " }{OLE 1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::
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{TEXT -1 1 " " }{XPPEDIT 18 0 "diff(``,y)" "6#-%%diffG6$%!G%\"yG" }
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{XPPEDIT 18 0 "diff(``,x,x)" "6#-%%diffG6%%!G%\"xGF'" }{TEXT -1 3 " + \+
" }{XPPEDIT 18 0 "diff(``,y,y)" "6#-%%diffG6%%!G%\"yGF'" }{TEXT -1 3 "
 + " }{XPPEDIT 18 0 "diff(``,z,z)" "6#-%%diffG6%%!G%\"zGF'" }{TEXT -1 
3 " ) " }{TEXT 367 18 "u                 " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 281 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "u[xx]+u[yy]+
u[zz]" "6#,(&%\"uG6#%#xxG\"\"\"&F%6#%#yyGF(&F%6#%#zzGF(" }{TEXT -1 28 
"                            " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 282 "" 0 "" {TEXT -1 3 " = " }{TEXT 369 1 "f" }{XPPEDIT 18 0 "``
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                              " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Thus, th
e laplacian operator is generally called \"del-squared\", and in carte
sian coordinates is written" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
262 "" 0 "" {OLE 1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyy
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{TEXT -1 3 " + " }{TEXT 385 1 "w" }{TEXT -1 1 " " }{TEXT 382 1 "k" }
{TEXT -1 2 ") " }{TEXT 379 1 "." }{TEXT -1 2 " (" }{TEXT 374 1 "i" }
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{TEXT -1 1 " " }{XPPEDIT 18 0 "diff(``,z)" "6#-%%diffG6$%!G%\"zG" }
{TEXT -1 2 ") " }}{PARA 285 "" 0 "" {TEXT -1 3 " = " }{TEXT 387 1 "u" 
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{TEXT -1 21 "                     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 63 "The directional derivative of Section 18.
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:4:" }{TEXT -1 248 "  is as defined in the first generalization of the
 directional derivative.  The formalism for the second generalization \+
is established in Exercise B1 of Section 18.2.  The idea behind the fo
rmalism is a generalization of the directional derivative." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Think of " }{TEXT 
413 1 "F" }{TEXT -1 26 " as a force field in which" }}{PARA 288 "" 0 "
" {TEXT 410 1 "R" }{XPPEDIT 18 0 "``(p)" "6#-%!G6#%\"pG" }{TEXT -1 3 "
 = " }{TEXT 411 1 "P" }{XPPEDIT 18 0 "``[1]+p" "6#,&&%!G6#\"\"\"F'%\"p
GF'" }{TEXT 412 1 "V" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 18 "is a line through " }{XPPEDIT 18 0 "P[1
]" "6#&%\"PG6#\"\"\"" }{TEXT -1 16 " with direction " }{TEXT 414 1 "V
" }{TEXT -1 37 ".  Evaluating F along the line gives " }{TEXT 415 1 "F
" }{XPPEDIT 18 0 "``(p)" "6#-%!G6#%\"pG" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "d/dp" "6#*&%\"dG\"\"\"%#dpG!\"\"" }{TEXT -1 1 " " }
{TEXT 416 1 "F" }{TEXT -1 2 " |" }{XPPEDIT 18 0 "``[p=0]" "6#&%!G6#/%
\"pG\"\"!" }{TEXT -1 56 " is the directional derivative of the field m
easured at " }{XPPEDIT 18 0 "P[1]" "6#&%\"PG6#\"\"\"" }{TEXT -1 21 " i
n the direction of " }{TEXT 417 1 "V" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "When this process was c
arried out in Section 18.2 for " }{TEXT 418 1 "D" }{XPPEDIT 419 0 "``[
U]" "6#&%!G6#%\"UG" }{TEXT -1 1 " " }{TEXT 420 1 "f" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "grad(f)" "6#-%%gradG6#%\"fG" }{TEXT -1 1 " " }{TEXT 
422 1 "." }{TEXT -1 1 " " }{TEXT 421 1 "U" }{TEXT -1 49 ", the directi
onal derivative of the scalar field " }{XPPEDIT 18 0 "f" "6#%\"fG" }
{TEXT -1 22 ", the gradient vector " }{XPPEDIT 18 0 "grad(f)" "6#-%%gr
adG6#%\"fG" }{TEXT -1 59 " appeared as a new object dotted with the di
rection vector " }{TEXT 423 1 "U" }{TEXT -1 144 ".  The matrix which a
ppears in the second generalization of the directional derivative is a
 new object, one \"rank\" higher that the vector field " }{TEXT 424 1 
"F" }{TEXT -1 75 ", called the covariant derivative.  Multiplication o
f the direction vector " }{TEXT 425 1 "V" }{TEXT -1 49 " by this objec
t gives the desired rate of change." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "
" {TEXT -1 21 "Example (Not in Text)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Let the vector belonging to the \+
operator be " }{TEXT 290 1 "V" }{TEXT -1 10 ", given by" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(V
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 39 "and let the operator act on the vector " }{TEXT 
426 1 "F" }{TEXT -1 10 ", given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := vector([f(x,y,z), g(x,
y,z), h(x,y,z)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "In Maple, the operator itself cann
ot be represented.  However, the " }{TEXT 291 6 "action" }{TEXT -1 31 
" of the operator on the vector " }{TEXT 428 1 "F" }{TEXT -1 43 " can \+
be computed with the following syntax." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "map(dotprod,map(grad, F
, [x,y,z]),V, orthogonal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Each of the scalar compone
nts of the vector " }{TEXT 427 1 "F" }{TEXT -1 81 " are subjected to t
he generalized directional derivative, yielding the components" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "for k from 1 to 3 do\ndotprod(V, grad(F[k],[x,y,z]), orthogonal)
;\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }