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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 279 0 "" }{TEXT 
280 23 "Calculus IV with Maple\n" }{TEXT 281 32 "Copyright 2002, Dr. J
ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 58 "\nLesson 2: Vector Derivatives: grad, div
, curl, laplacian\n" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 0 "" }{TEXT 257 0 "" }{TEXT 258 128 "Topics : Gradient, divergence
, curl are illustrated and graphically interpreted. Laplacian defined.
 \nMaple commands introduced: " }{TEXT 259 56 "Gradient, Divergence, C
url, Laplacian, fieldplot, evalVF" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Gradient" 2 "Gradient" "
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
260 11 "Example 2.1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Let  " }
{XPPEDIT 18 0 "f = sin(x)-log(sin(y));" "6#/%\"fG,&-%$sinG6#%\"xG\"\"
\"-%$logG6#-F'6#%\"yG!\"\"" }{TEXT -1 54 ".  Compute and plot grad w. \+
  We will plot level sets " }{XPPEDIT 18 0 "w^(-1);" "6#)%\"wG,$\"\"\"
!\"\"" }{TEXT -1 1 "(" }{TEXT 266 1 "k" }{TEXT -1 26 ") , for k from 0
.1 to 1.0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with
(plots): with(LinearAlgebra): with(VectorCalculus):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "f := sin(x) - log(sin(y));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Plotting \+
ten level sets of " }{TEXT 287 7 "f(x, y)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "S := seq(f - 1/k = 0, k = 1..10):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "P1 := implicitplot(\{S\}, x = 3..3.5, y =
 0..1.5, numpoints = 1000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "display(P1, scaling = constrained); \n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "The value of k represented by these curves increases from
 right to left. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 56 "Now compute the gradient at x = 3.3 on the level s
et of " }{XPPEDIT 18 0 "f(x,y)=1/10" "6#/-%\"fG6$%\"xG%\"yG*&\"\"\"F*
\"#5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gf := Gradient
(f - 1/10, [x, y]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Y \+
:= fsolve(subs(x = 3.3, f - 1/10), y);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "G := evalVF(gf, <3.3, Y>);" }}}{EXCHG {PARA 0 "" 0 "
" {HYPERLNK 17 "evalVF" 2 "evalVF" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Now we dr
aw and plot a line through the point [3.3, .8832282346], and parallel \+
to G." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N := <3.3, Y> + s \+
* G;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "P2 := plot([N[1],
 N[2], s =  - 0.2..0.3], thickness = 2, color = black):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "display(P1, P2, scaling = constrain
ed);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "The gradient vector ap
pears to pierce each of the level sets orthogonally to its tangent.  I
f this is so, then the gradient vector represents  the shortest distan
ce between successive level sets and, therefore, the most rapid change
 in the value of w. This is, in fact, true of any function in any dime
nsion." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We consider a surface o
f arbitrary dimension represented by a level set of a map:  F: " }
{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 3 " ->" }{TEXT 267 1 "R
" }{TEXT -1 4 " ;  " }{TEXT 268 1 "f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "
x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 3 "..." }{XPPEDIT 18 0 "x[n];" "6#
&%\"xG6#%\"nG" }{TEXT -1 6 ")  =  " }{TEXT 269 1 "K" }{TEXT -1 15 ", a
nd a curve, " }{XPPEDIT 18 0 "gamma(t) = [x[1](t) .. x[n](t)];" "6#/-%
&gammaG6#%\"tG7#;-&%\"xG6#\"\"\"6#F'-&F,6#%\"nG6#F'" }{TEXT -1 62 "   \+
 , imbedded in the surface, and passing through a point,   " }
{XPPEDIT 18 0 "p = x[1](t[0]) .. x[n](t[0]);" "6#/%\"pG;-&%\"xG6#\"\"
\"6#&%\"tG6#\"\"!-&F(6#%\"nG6#&F-6#F/" }{TEXT -1 7 " .     " }
{XPPEDIT 18 0 "F(gamma(t)) = f(x[1](t) .. x[n](t));" "6#/-%\"FG6#-%&ga
mmaG6#%\"tG-%\"fG6#;-&%\"xG6#\"\"\"6#F*-&F16#%\"nG6#F*" }{TEXT -1 3 " \+
= " }{TEXT 270 1 "K" }{TEXT -1 25 "   Taking the gradient:  " }
{XPPEDIT 18 0 "grad(F(gamma(t)));" "6#-%%gradG6#-%\"FG6#-%&gammaG6#%\"
tG" }{TEXT -1 12 "  =  grad F(" }{XPPEDIT 18 0 "gamma(t);" "6#-%&gamma
G6#%\"tG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT 
-1 2 "'(" }{TEXT 271 1 "t" }{TEXT -1 16 ")  =  0.  Since " }{OLE 1 
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya
:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z
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xYgB@nAlj:F:=b:?b:?B:?B:=R:=j;@j:^:dj:<J;LJ<Iu;XP;LJ;Y\\aNZ>BD\\L;Lj;M
anOZ>NjcRD<j?:^;yayI:>:>\\:B:qQBv:<J:^q<B:;B:=b:Dlc`qsLqlp@CZ:f?=:>X;j
;<Z:>::::::::::sB:;B:?JsXXtKPQYwJ:n@KM>QYwfZ;:RG<:::E=:t@:VDD`f;J]ZokB
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<b:?B:F:V:^:dZ::ZaZ>BDj;K;RDJtDA<:::::::::::::::::::::::::::::::JhB:;:
;:::::::3:" }{TEXT -1 48 " is the principal part of the tangent vector
 to " }{OLE 1 4160 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::
yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=
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J@j@>:W:YJ:><<JyKC>:a:c:e:wAyA::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::=Z<Z::::::::::j:nyyYZDjysyAbm?:;JZC:bKg:
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<:=:A:CZDB::BDj;K;RDJ:<:::::::::::::::::::::::::::::::::::::::::5:" }
{TEXT -1 8 ",  grad " }{TEXT 272 1 "F" }{TEXT -1 1 "(" }{XPPEDIT 18 0 
"gamma;" "6#%&gammaG" }{TEXT -1 1 "(" }{TEXT 273 1 "t" }{TEXT -1 46 ")
)  is orthogonal to the tangent to the curve " }{XPPEDIT 18 0 "gamma(t
);" "6#-%&gammaG6#%\"tG" }{TEXT -1 155 " at p.  But every tangent to t
he surface at p, is the tangent to a curve,  (existence theorem for so
lution of first order differential equations) so  grad " }{TEXT 274 1 
"F" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 1 "
(" }{TEXT 275 1 "t" }{TEXT -1 170 "))  is orthogonal to the tangent pl
ane at p.  The gradient vector represents the direction and magnitude \+
of the greatest change in F, at the point where it is evaluated. " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Th
is can be illustrated easily in three dimensions. " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Example 2.2" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Le
t  " }{OLE 1 4168 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::y
yyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
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::::::::::::::::::::::::::::::::::::::::::::5:" }{TEXT -1 108 " .  Com
pute and plot the gradient vector.\nDefine an implicit function of (x,
 y, z) for various values of K.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "restart: with(plots): with(LinearAlgebra):with(Vector
Calculus): with(plottools):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "F := K ->z - x^2 + y^2 + 4 * x * y - K:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "v := [x, y, z]: gradF := Gradient(F(0), v);\n" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 
"We could equally well have taken any value of K. 0 happens to be conv
enient  because it gives simple integer values for the components of t
he gradient." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p := <1, 1,
  - 4>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "gF[1, 1,  - 4] \+
:= evalVF(gradF, p);  #The gradient vector at (1, 1,  - 4)\n          \+
                               " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "G := p + s * gF[1, 1,  - 4]; \011\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 88 "P1 := spacecurve(evalm(G), s =  - .2..0.25, color \+
= black, thickness = 2, axes = boxed):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 139 "S := seq(implicitplot3d(F(5 * k), x = 0..2, y = 0..2
, \nz =  - 5.. - 2, style = wireframe, color = pink, grid  =  [12, 12,
 12]), k = 0..3):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "disp
lay(S, P1, scaling = constrained, orientation = [ - 20, 65]);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The gradient vector appears to be \+
orthogonal to each of the level sets.\n                    " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 277 "Without presenting any mathematic
al justification (although ample justification exists), we will operat
e with the gradient vector operator as if it were an ordinary vector; \+
we will take both the dot product and the  crossproduct of  del with o
rdinary vectors and with itself.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 " The curl of a vector V is defined by: curl(V) =  del" }{TEXT 
276 1 "x" }{TEXT -1 1 "V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 
"restart: with(LinearAlgebra): with(VectorCalculus):with(linalg):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Th
ere are two defined forms of the curl in Maple. One, spelled with a sm
all \"c\" is in the " }{TEXT 282 6 "linalg" }{TEXT -1 82 " package and
 operates on vectors; the other, spelled with a capital \"C\" is in th
e " }{TEXT 283 14 "VectorCalculus" }{TEXT -1 39 " package and operates
 on vector fields." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "curl" 2 "c
url" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Cur
l" 2 "Curl" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "V := VectorField(<v[1](x, y, z), v[2](x, y, z), v[3](
x, y, z)>, 'cartesian'[x, y, z]); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "Curl(V);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "V1 := [v[1](x, y, z), v[2](x, y, z), v[3](x, y, z)];r := [x, y, z]
:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "curl(V1, r);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "T
o have an idea of what this means, consider a vector field, that is, a
 function that assigns a vector to each point in the domain." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 11 "E
xample 2.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 " Let  " }{TEXT 277 0 "" }{XPPEDIT 18 0 "F(x,y) = [x^2*y, y
^2*x, x^2-y^2];" "6#/-%\"FG6$%\"xG%\"yG7%*&F'\"\"#F(\"\"\"*&F(F+F'F,,&
*$F'F+F,*$F(F+!\"\"" }{TEXT -1 61 "      Take the curl of this vector \+
field and fieldplot both. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {HYPERLNK 17 "fieldplot" 2 "fieldplot" "" }{MPLTEXT 1 
0 2 "  " }{HYPERLNK 17 "fieldplot3d" 2 "fieldplot3d" "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "restart: with(LinearAlgebra): with(
VectorCalculus):with(linalg):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "F := (x, y) -><x^2 * y, y^2 * x, x^2 - y^2>:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(x, y);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "curl_F := curl(F(x, y), [x, y, z]);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "f1 := fieldplot3d(F(x, y), \+
x =  - 10..10, y =  - 10..10, z = \n - 10..10, axes = framed, color = \+
red, arrows = THICK, grid = [3, 3, 3]):\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 137 "f2 := fieldplot3d(curl_F, x =  - 10..10, y =  - 10
..10, \nz =  - 10..10, axes = framed, color = black, arrows = THICK, g
rid = [3, 3, 3]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "disp
lay(f1, f2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 502 "If the origina
l vector field F, (thick arrows) represents the velocity of fluid flow
 in a container, then curl F (the thinner arrows) is the torque exerte
d by the fluid, both in magnitude and direction, (determined by the ri
ght hand rule).  If  any group of thick vectors is rotated into an adj
acent group and the fingers of the right hand curled in the direction \+
of rotation, the thumb of the right hand will point in the direction o
f the thinner vectors (the torque) situated between the two groups. " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 
"Now formally defining the vector field." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "FF := VectorField(F(x, y), 'cartesian'[x, y, z]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "CFF := Curl(FF);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "P3 := fieldplot3d(FF, x =  \+
- 10..10, y =  - 10..10, z = \n - 10..10, axes = framed, color = red, \+
arrows = THICK, grid = [3, 3, 3]):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "P4 := fieldplot3d(CFF, x =  - 10..10, y =  - 10..10,
 \nz =  - 10..10, axes = framed, color = black, arrows = THICK, grid =
 [3, 3, 3]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display(P
3, P4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The alternative to " }
{TEXT 286 4 "Curl" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "Del &x FF;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 69 "Note that, because of the equality of mi
xed partial derivatives,  del" }{TEXT 278 1 "x" }{TEXT -1 9 "del = 0. \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "del := <d/dx, d/dy, d/d
z>; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "del &x del;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 290 "Since we are considering del as a vector we may also take the \+
dot product of del with a vector, V.  This is known as the divergence \+
of V.  Again there are two definitions, Divergence and diverge in the \+
VectorCalculus and linalg packages, operating on vector fields and vec
tors respectively." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Divergence
" 2 "Divergence" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{HYPERLNK 17 "diverge" 2 "diverge" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "restart: with(linalg): with(LinearA
lgebra): with(VectorCalculus): with(plots):\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 65 "V := (x, y, z) -> <v[1](x, y, z), v[2](x, y, z),
 v[3](x, y, z)>: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "VV := \+
VectorField(V(x, y, z), 'cartesian'[x, y, z]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "Divergence(VV);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "V1 := [v[1](x, y, z), v[2](x, y, z), v[3](x, y, z)];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diverge(V1, [x, y, z]);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
215 "To have a picture of what this might mean, suppose we consider F \+
as the velocity field of some gas.  That is, F is a vector field repre
senting the velocity of the gas particles (assumed to be uniform) at e
ach point." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 263 11 "Example 2.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 32 "Find and plot the divergence of " }{OLE 
1 4164 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::
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@Jn@jk@Jb@jFHJ:Hj<<:c:CY:c:qAB:>L;B:;R:D:ETWAEUMES;sNmMkxPpFZ:^:f?=J<J
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qjN<KRBYSKYFKYSJLYBxj]xoZ<OsyKI@NbLyDxjZ<JtQNtQLtyR[xoZB>Hni@>YVKciDyD
Lcy]=D:<[:_xIHby?cnA\\IFbdve>we?_xYgB@nAlj:F:=b:?b:?B:?j:@j:VZ;FZ:^:dj
:<J;Lj;U^@OZ>NjtbH?B;GVIpM;LJ;i_UOZ>VjrCXAJH>ZEFZ;F:?J>E]lJ?DJ<HrvFi<^
Z<`?Sc:CrZr:l;::ZrK;LJ;m<L:V::::TRCDlPDJ<Hrv::::?VQp?QB::Sjysy=Z:>:>\\
:B:qQBv:>:sg:<J:<j:DZ\\VdsWhngfgK<B:UK:^:>X=B:AB:<J:::::::::::::::::::
:::5:" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "re
start: with(plots): with(LinearAlgebra): with(VectorCalculus):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F := (x, y) ->[x^2 * y, y^2 \+
* x, x^2 - y^2]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "FF :=
 VectorField(<F(x, y)>, 'cartesian'[x, y, z]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "Div_F := Divergence(FF);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 197 "The divergence is a scalar quantity.  If we const
ruct a vector parametrized by x and y with div_F as the z coordinate, \+
we will obtain a surface representing div F  at each point in the x - \+
y plane." }{MPLTEXT 1 0 1 " " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "G := <x, y, 4 * x * y>;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 30 "We now plot G together with F." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 90 "P1 := plot3d(G, x =  - 2..2, y =  - 2..2, axes
 = framed, color = blue, style = wireframe):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 144 "P2 := fieldplot3d(F(x, y), x =  - 2..2, y =  - \+
2..2, z = \n - 2..2, axes = framed, color = pink, color = red, arrows \+
= THICK, grid = [5, 5, 2]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "display(P1, P2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 382 " The
 surface div V represents the flux of gas out of a unit cube.  Observe
 that where the arrows point inward, the flux is negative,  gas partic
les accumulate in the cube; the density of the gas rises.  Where the a
rrows point outward, the flux is positive, the density of gas particle
s in the cube is diminishing. div F therefore measures the change in d
ensity of gas in the cube.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 83 "div(grad) is defined and is an operator
 of the greatest importance: the Laplacian. " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 66 "Maple does not have a way to input partial differential
 operators." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "del := <d/dx
, d/dy, d/dz>; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Lap := d
el.del;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The Laplacian functi
on, is however, defined in the " }{TEXT 284 6 "linalg" }{TEXT -1 17 " \+
package and the " }{TEXT 285 14 "VectorCalculus" }{TEXT -1 9 " package
." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "laplacian" 2 "laplacian" "
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Laplacian
" 2 "Laplacian" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Grad, curl and div figure larg
ely in physical applications and will all be looked at more closely as
 we proceed. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "
" }{TEXT 265 8 "Practice" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "1. Plot each of the following vect
or fields together with its curl." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
4 "a.  " }{XPPEDIT 18 0 "F(x,y) = [x+y, x-y, x*y];" "6#/-%\"FG6$%\"xG%
\"yG7%,&F'\"\"\"F(F+,&F'F+F(!\"\"*&F'F+F(F+" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 3 "b. " }{XPPEDIT 18 0 "F(x,y) = [-x/(x^2+y^2), y/(x^2+y^2)
, 1/(x^2+y^2)];" "6#/-%\"FG6$%\"xG%\"yG7%,$*&F'\"\"\",&*$F'\"\"#F,*$F(
F/F,!\"\"F1*&F(F,,&*$F'F/F,*$F(F/F,F1*&F,F,,&*$F'F/F,*$F(F/F,F1" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "c. " }{XPPEDIT 18 
0 "F(x,y) = [sin(x+y), cos(x+y), x+y];" "6#/-%\"FG6$%\"xG%\"yG7%-%$sin
G6#,&F'\"\"\"F(F.-%$cosG6#,&F'F.F(F.,&F'F.F(F." }{TEXT -1 0 "" }}}}
{MARK "0 0 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }