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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 43 "The Gradient and the Dir
ectional Derivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 34 "The next cell shows how the Maple " }{TEXT 258 6 "plot3
d" }{TEXT -1 44 " procedure can be used to study a surface.  " }{TEXT 
256 15 "Evaluate it now" }{TEXT -1 257 ".  This particular function ca
n be hard to visualize.  Try looking at the graph from various differe
nt perspectives by clicking on the graph to activate it and then dragg
ing the bounding box o look at the surface from various different pers
pectives.  Press " }{TEXT 257 5 "enter" }{TEXT -1 40 " each time you w
ant to redraw the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := (x, y) -> x * y:\n\nplot3d(f(x
, y), x=-1..1, y=-1..1, scaling=constrained, axes=boxed);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "The next
 cell draws a graph similar to the one in the browser window shoing wh
at happens as you walk one unit in various different directions starti
ng from the same point.  " }{TEXT 259 15 "Evaluate it now" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1099 "with(plots):\n\nx0 := 0:\ny0 := 0:\n\nsurface := pl
ot3d(f(x, y), x=-1..1, y=-1..1, \n                 scaling=constrained
, axes=boxed, style=patchnogrid):\n\ncirc := spacecurve([x0 + cos(t), \+
y0 + sin(t), f(x0 + cos(t), y0 + sin(t))],\n                  t=0..2 *
 Pi, color=black, thickness = 2,\n                  numpoints=200):\n
\nu1 := [1/sqrt(2), 1/sqrt(2)]:\nu2 := [1/sqrt(2), -1/sqrt(2)]:\nu3 :=
 [-1/sqrt(2), 1/sqrt(2)]:\nu4 := [-1/sqrt(2), -1/sqrt(2)]:\n\n\ncurve1
 := \n   spacecurve(\n       [x0 + t * u1[1], y0 + t * u1[2], f(x0 + t
 *u1[1], y0 + t * u1[2])],\n              t=0..1, color=red, thickness
=2):\n\ncurve2 := \n   spacecurve(\n       [x0 + t * u2[1], y0 + t * u
2[2], f(x0 + t *u2[1], y0 + t * u2[2])],\n              t=0..1, color=
red, thickness=2):\n\ncurve3 := \n   spacecurve(\n       [x0 + t * u3[
1], y0 + t * u3[2], f(x0 + t *u3[1], y0 + t * u3[2])],\n              \+
t=0..1, color=red, thickness=2):\n\ncurve4 := \n   spacecurve(\n      \+
 [x0 + t * u4[1], y0 + t * u4[2], f(x0 + t *u4[1], y0 + t * u4[2])],\n
              t=0..1, color=red, thickness=2):\n\ndisplay(\{surface, c
irc, curve1, curve2, curve3, curve4\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The next cell plots two c
urves like the curves drawn above but in two-dimensions . " }{TEXT 
262 15 "Evaluate it now" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 91 "plot(\{f(x0 + t * u1[1], y0 + t * u1[2]),\n      f(
x0 + t * u2[1], y0 + t * u2[2])\}, t=0..1);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 79 "The next cell draws a vector field showing the gradi
ent vector at each point.  " }{TEXT 260 15 "Evaluate it now" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 194 "f := (x, y) -> x^2 - y^2:\n\nfx := (x, y) -> diff(f(
x, y), x):\nfy := (x, y) -> diff(f(x, y), y):\n\nfieldplot([fx(x, y), \+
fy(x, y)], x=-1..1, y=-1..1, scaling=constrained,\n          grid = [1
0, 10]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 0" 0 }
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