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olya vector field F of a function f(z)=f(x+iy)=u(x,y)+i*v(x,y) is defi
ned to be the set of vectors" }}{PARA 256 "" 0 "" {TEXT -1 95 "F=[F_x,
F_y]=[u(x,y),-v(x,y)] in the plane.  Here F_x, and F_y, denote the x a
nd y components of" }}{PARA 256 "" 0 "" {TEXT -1 24 "the vector field.
 (They " }{TEXT 267 6 "do not" }{TEXT -1 62 " represent partial deriva
tives.)  In the theory of electricity" }}{PARA 256 "" 0 "" {TEXT -1 
65 "and magnetism, one can think of the Polya vector field as either \+
" }}{PARA 256 "" 0 "" {TEXT -1 47 "an electrostatic field F or a magne
tic field B." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 96 "Below is the Polya vector field corresponding to f(z)=1/z
. Note the behavior of the field at the" }}{PARA 256 "" 0 "" {TEXT -1 
78 "singularity z=0. In physics, the singularity at z=0 represents a p
oint charge." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=z->z^2;
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0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 95 "
1. Where are the arrows largest in length? Where are they smallest in \+
length. Through how many " }}{PARA 0 "" 0 "" {TEXT 269 73 "revolutions
 do the arrows rotate as one travels a path around the origin?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 270 62 "2. Repeat the above question using f
(z)=z^2 and then f(z)=z^3." }}{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 -1 65 "The DEL opera
tor is defined to be the partial derivative operator" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(i,x);" "6#-%%d
iffG6$%\"iG%\"xG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(j,y);" "6#-%%
diffG6$%\"jG%\"yG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(k,z);" "6#-%
%diffG6$%\"kG%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 256 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "divergence" }
{TEXT -1 66 " of a vector field F = [F_x, F_y] at a point is defined t
o be the " }}{PARA 256 "" 0 "" {TEXT 256 6 "scalar" }{TEXT -1 33 " qua
ntity obtained by taking the " }{TEXT 258 11 "dot product" }{TEXT -1 
30 " of DEL with the value of the " }}{PARA 256 "" 0 "" {TEXT -1 40 "v
ector F at that point. In other words, " }}{PARA 256 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "div(F) = " }{XPPEDIT 18 0 "diff(F
_x,x);" "6#-%%diffG6$%$F_xG%\"xG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "di
ff(F_y,y);" "6#-%%diffG6$%$F_yG%\"yG" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 59 "Intuitively, the d
ivergence measures net inflow/outflow of " }}{PARA 256 "" 0 "" {TEXT 
-1 34 "the vector field at that point.   " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 94 "3. Show that if f is differentia
ble at a point z=z0, then the divergence of the corresponding " }}
{PARA 256 "" 0 "" {TEXT -1 22 "Polya vector field is " }{TEXT 263 11 "
zero there." }{TEXT -1 73 " (Hint: Use the dot product, together with \+
the Cauchy-Riemann equations.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 64 "Below Ma
ple calculates the divergence for the Polya field above." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diverge([op(F),0], [x,y,z]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"\"F%,&*$)%\"xG\"\"#F%F%*$)%\"y
GF*F%F%!\"\"F**&*&F*F%F(F%F%*$)F&F*F%F.F.*&*&F*F%F,F%F%*$F2F%F.F." }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 4 "curl" }{TEXT 
-1 65 " of a vector field F = [F_x, F_y] at a point is defined to be t
he" }}{PARA 256 "" 0 "" {TEXT 257 6 "vector" }{TEXT -1 29 " quantity o
btained by taking " }{TEXT 261 13 "cross product" }{TEXT -1 20 " of th
e DEL operator" }}{PARA 256 "" 0 "" {TEXT -1 52 "with the vector F_z*i
 + F_y*j + F_z*k at that point." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 78 "Intuitively, the curl measures rotation
al tendency of the vector field at that" }}{PARA 256 "" 0 "" {TEXT -1 
85 "point. Think of placing a paddle wheel placed at a point with its \+
blade perpendicular" }}{PARA 0 "" 0 "" {TEXT 262 75 "to the plane. If \+
the paddle wheel rotates, the curl is different from zero." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 88 "4. Show that if f is differentiable at a point z=z0,
 then the curl of the corresponding " }}{PARA 256 "" 0 "" {TEXT -1 22 
"Polya vector field is " }{TEXT 265 11 "zero there." }{TEXT -1 76 "  (
Hint: Use the cross product, together with the Cauchy-Riemann equation
s.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 58 "Below M
aple calculates the curl for the Polya field above." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "curl([op(F),0], [x,y,z]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 256 "" 0 
"" {TEXT -1 91 "The above results have a significant connection to phy
sics, as the curl and divergence are " }}{PARA 256 "" 0 "" {TEXT -1 
89 "essential for formulating the differential form of Maxwell's equat
ions, the four primary " }}{PARA 256 "" 0 "" {TEXT -1 60 "equations th
at govern the theory electricity and magnetism. " }}}}{MARK "11 1 1" 
17 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }