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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "What would you see if you poured
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0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "The water would f
low down the inverted paraboloid and if we view all of this from direc
tly above, we see that the water flows along the arrows shown." }}
{PARA 0 "" 0 "" {TEXT -1 344 "The gradient vector field represents the
 directions along which the potential is increasing the fastest, and s
o the negative of the gradient represents the directions along which t
he potential is decreasing the fastest--the direction the water would \+
follow.  Water would flow from the highest point to the lowest under t
he influence of gravity." }}{PARA 0 "" 0 "" {TEXT -1 308 "If we plot t
he gradient vector field, we invert the arrows, or turn gravity upside
 down; this is a bit awkward, so we will stick to plotting the negativ
e of the gradient and continue to use this \"pouring water on the grap
h of a potential function\" idea to talk about potential, gradient, cu
rl and divergence." }}{PARA 0 "" 0 "" {TEXT -1 59 "Notice that the div
ergence is non-zero and the curl is zero" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "diverge([x,y,0],[x,y,z]);" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }{XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "curl([x,y,0],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec
torG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Now consider
 the potential " }{XPPEDIT 18 0 "(x^2+y^2)/2;" "6#*&,&*$%\"xG\"\"#\"\"
\"*$%\"yGF'F(F(F'!\"\"" }{TEXT -1 19 " and plot its graph" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot3d((x^2+y^2)/2,x=-2..2,y=-2..2)
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2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "All the arrows point towards the \+
center and so it must be a sink.  In the previous example, water was f
lowing down the paraboloid and hence moving away from the origin in th
e negative gradient field plot--that was a source." }}{PARA 0 "" 0 "" 
{TEXT -1 44 "Again notice that the divergence is non-zero" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "diverge([-x,-y,0],[x,y,z]);" }{TEXT -1 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 431 "The divergence measures the presence of sinks and source
s, and in general, measures the total fluid flow across any closed cur
ve--the diference between the total fluid moving out and the total flu
id coming in.  If there's a sink, total fluid flow should be negative \+
across any closed curve enclosing that sink only, and if there's a sou
rce the total fluid flow should be positive across any closed curve en
closing that source only." }}{PARA 0 "" 0 "" {TEXT -1 136 "Here we are
 measuring the divergence of the negative gradient field; for the dive
rgence of the gradient vector field, reverse the signs." }}{PARA 0 "" 
0 "" {TEXT -1 33 "Note that the curl is still zero." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "curl([-x,-y,0],[x,y,z]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 616 "Level curves are also another way to view potential func
tions.  In the above paraboloid--the graph of the potential function--
the circles represent points that are at the same height (potential). \+
 If we project these down to the plane, we get the level curves, curve
s on which the potential is constant.  These level curves are orthogon
al to the gradient (or the negative gradient) vector field.  This is b
ecause water will find the shortest/most efficient path to flow from h
igher points to lower points; it will not stay at the same height but \+
flow down in the direction perpendicular to points at the same height.
" }}{PARA 0 "" 0 "" {TEXT -1 277 "Becuase the level curves are orthogo
nal to the gradient vector field, we can easily find them by rotating \+
the gradient (or the negative gradient) vector field by 90 degrees.  T
his rotated vector field is called the Hamiltonian vector field, and i
ts orbits are the level curves." }}{PARA 0 "" 0 "" {TEXT -1 63 "So in \+
the above example the Hamiltonian vector field is <y,-x>." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 58 "fieldplot([y,-x],x=-2..2,y=-2..2,arrows=SLIM
,color=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 449 451 451 {PLOTDATA 2 "
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\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The divergence shouldn'
t be zero, because it should detect the presence of the sinks and sour
ces." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diverge([-cos(x)*cos(y),sin
(x)*sin(y),0],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"
\"\"-%$sinG6#%\"xGF&-%$cosG6#%\"yGF&F&" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 300 "It can be easily checked that the divergence is positive
 at the sources and negative at the sinks.  Furthermore, in this case,
 the divergence is exactly zero at the saddles.  Not all saddles get a
way with contributing zero divergence, but some special ones do.  Whic
h ones contribute zero divergence?" }}{PARA 0 "" 0 "" {TEXT -1 24 "Aga
in, the curl is zero." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "curl([-cos
(x)*cos(y),sin(x)*sin(y),0],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "Th
e curl has been zero in all the examples so far, and this is no coinci
dence.  The vector fields that we have been taking the curl of have al
l been (negative) gradient fields and by the famous theorem curl(grad)
 = 0, we should have obtained zero." }}{PARA 0 "" 0 "" {TEXT -1 413 "W
hy is curl(grad) = 0?  Curl measures the vorticity or the rotation of \+
the fluid flow.  If fluid is flowing down a graph of a potential funct
ion is it rotating while doing so?  In examples above, the water seems
 to flow straight down without any rotation, and hence, the curl is ze
ro in all these examples.  But does this always happen?  Does water al
ways flow down the graph of a potential without any vorticity?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Let's ex
amine some examples that do have curl.  The simplest one comes from th
e vector field <-y, x>." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "
fieldplot([-y, x],x=-4..4,y=-4..4,arrows=SLIM,color=blue);" }}{PARA 
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"Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This vector field ce
rtainly possesses rotation and its curl is non-zero." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "curl([-y,x,0],[x,y,z]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 452 "Does this vector field arise as a gradient of some pot
ential?  If it does the fluid must flow along the arrows in closed cir
cular orbits.  But water flows from a higher potential to a lower pote
ntial, from higher points to lower points under gravity.  If the orbit
s are circular, water must flow down and then flow up to come back aga
in, flowing against gravity.  This is impossible!  So the vector field
 does not arise as a gradient of some potential." }}{PARA 0 "" 0 "" 
{TEXT -1 186 "Again, this shouldn't be surprising.  A neccessary (and \+
sufficient) condition for a vector field to be the gradient of some po
tential function in the plane is that its curl must be zero." }}{PARA 
0 "" 0 "" {TEXT -1 41 "Note that the divergence of curl is zero." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diverge([0,0,2],[x,y,z]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
42 "Here's another example with non-zero curl." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "fieldplot([-y+x, x+y],x=-4..4,y=-4..4,arrows=S
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "The orbits all spiral out from the origin
--these are the level curves." }}{PARA 0 "" 0 "" {TEXT -1 665 "The lev
el curves represent constant potentials or constant heights on the gra
ph of the potential function.  If two level curves intersect at a poin
t, then every point on both curves must be at the same potential (heig
ht).  Above, all the level curves are spirals with a common intersecti
on point at the origin.  So every point is at the same potential and t
he graph of the potential function must be flat--everything is at the \+
same height.  But water doesn't flow anywhere on a flat surface, it ju
st sits there.  However, we have a non-stationary vector field.  This \+
is impossible!  So again, it couldn't have been obtained as a gradient
 of some potential function." }}{PARA 0 "" 0 "" {TEXT -1 172 "This lev
el curve discussion should be considered as the modified \"flowing aga
inst gravity\" argument; it's more general and could have been used in
 the earlier example also." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 602 " These examples suggest that if the curl is non-
zero, the vector field is rotating about some point, and then the leve
l curves show that the vector field cannot have arisen as the gradient
 of some potential function.  Also, the curl of a gradient vector fiel
d is zero since water flows down the graph of a potential function wit
hout rotating--if it did, curl would be non-zero and that would imply \+
that the gradient vector field could not be a gradient vector field of
 some potential function!  So curl (grad) = 0 and if curl is non-zero,
 the vector field is not the gradient of any potential function." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Even in the second non-zero curl e
xample, notice that divergence of curl is zero" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "diverge([0,0,2],[x,y,z]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This is a
nother theorem, div (curl) = 0." }}{PARA 0 "" 0 "" {TEXT -1 179 "The t
rivial case is when curl is already zero, which may happen in the case
 of a gradient vector field.  The general case is not hard to visualiz
e for 2-dimensional vector fields." }}{PARA 0 "" 0 "" {TEXT -1 281 "Fo
r a 2-dimensional vector field, by the definition, curl looks like <0,
 0, f(x,y)>.  Curl is a 3-dimensional vector field, and in this case t
he first two components are zero with the last component a function of
 x and y.  So there is only one degree of freedom in the z direction.
" }}{PARA 0 "" 0 "" {TEXT -1 227 "Divergence detects the presence of s
inks and sources.  Both these require orbits coming in or going out fr
om a point in all directions.  A single degree of freedom cannot creat
e sinks or sources, and so divergence must be zero." }}{PARA 0 "" 0 "
" {TEXT -1 111 "A complete proof can be easily obtained by applying th
e formulas--it depends on the mixed partials being equal." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 483 "Getting y
our hands dirty with a few examples is never an exercise in futility. \+
 Playing with examples and drawing pictures helps build intuition and \+
reinforces the theorems proved by applying formulas, which is absolute
ly necessary for rigor.  We must make sure that the principles we thin
k that are true, by observing examples, are in fact true by rigorously
 checking them with sound mathematics.  For a more rigorous treatment \+
of these concepts, consult your local calculus book.  " }}}}{MARK "0 0
 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }