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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 4 "CURL" }{TEXT -1 1 "\n" }
{TEXT 257 18 "by Corinne Manogue" }}{PARA 0 "" 0 "" {TEXT -1 30 "Copyr
ight 2008 Corinne Manogue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 131 "In this work sheet you will look at some vector f
ields and calculate their curls, to get a feel for what the curl is ge
ometrically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 129 "restart:with(linalg):with(plots):\nsetoptions3d(scal
ing=constrained,grid=[10,10,10],\norientation=[-90,0],axes=framed,colo
r=black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "Define and plot som
e vectors in 2-dimensions and calculate the curl of the vector.  Note:
  even though we are interested in looking at a 2-dimensional vector, \+
we will define it to be a three component vector, where the z-componen
t is zero.  This is done because the curl operation \nis expecting a 3
-component vector." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "v1:=[0,1,0];
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Plot the vector field in \"2-
d\"--the third dimension is trivial." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "fieldplot3d(v1,x=-1..1,y=-1..1,z=-1..1,\narrows=SLIM);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "(Warning:  arrows are scaled so t
hat the longest arrow in a given graph \nhas length equal to the grid \+
unit)." }}{PARA 0 "" 0 "" {TEXT -1 84 "\nYou can calculate the curl of
 this function in your head...do you agree with Maple?" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "f1:=curl(v1,[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 29 "Now try another vector field:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "v2:=[x, y,0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "fieldplot3d(v2,x=-1..1,y=-1..1,z=-1..1,\narrows=SLIM);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "f2:=curl(v2,[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now \+
try another vector field:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v3:=[0
, exp(-y^2),0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "fieldplo
t3d(v3,x=-1..1,y=-1..1,z=-1..1,arrows=SLIM);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "f3:=curl(v3,[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 29 "Now try another vector field:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "v4:=[0, exp(-x^2),0];" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 444 "Imagine putting a paddle wheel in the plane of the plot,
 it will turn if the curl is non-zero.  If you put a paddle wheel in t
he center, what will happen? Where is the curl positive.  Where is it \+
negative?  (Recall, in a right handed coordinate system, you can use t
he right hand rule to determine which direction is positive.  Is posit
ive into or out of the page?)  Where is the curl zero?  Which vector c
omponent of the curl are we examining?" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 52 "fieldplot3d(v4,x=-1..1,y=-1..1,z=-1..1,arrows=SLIM);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f4:=curl(v4,[x,y,z]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now try another vector field:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v5:=[-y, x,0];" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 52 "fieldplot3d(v5,x=-1..1,y=-1..1,z=-1..1,arrow
s=SLIM);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f5:=curl(v5,[x,
y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now try another vector \+
field:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v6:=[-y/(x^2+y^2)
,x/(x^2+y^2),0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "fieldpl
ot3d(v6,x=-1..1,y=-1..1,z=-1..1,arrows=SLIM);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "f6:=map(normal,curl(v6,[x,y,z]));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "Now define a \"2-d\" vector field of your
 own, plot it, and find its curl." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Define and plot some vectors
 in 3 dimensions.  Calculate the curl of these vector fields:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "v7:=[x,y,z];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 77 "fieldplot3d(v7,x=-5..5,y=-5..5,z=-5..5,grid=[4
,4,4],axes=framed,arrows=SLIM);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
229 "By changing the orientation of the plot, you can view the plot in
 the plane of the coordinate axis (ie x-y plane or x-z plane or y-z pl
ane).  Convince yourself that each component of the curl is zero.  \nN
ow, calculate the curl.  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f7:=ma
p(normal,curl(v6,[x,y,z]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "No
w choose your own 3-d vector field, plot it, and find its curl:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }