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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 43 "Functions of Several Var
iables\nOctober 2008" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 26 "We will use the function  " }{XPPEDIT 18 0 "f(x,y
) = 4*x/(x^2+y^2+1);" "6#/-%\"fG6$%\"xG%\"yG*(\"\"%\"\"\"F'F+,(*$F'\"
\"#F+*$F(F.F+F+F+!\"\"" }{TEXT -1 68 "   in the computations below, an
d we define it now once and for all." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "f := (x,y) -> 4*x/(x^2+y^2 +1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 115 "Once we define f in the function syntax (with \" -> \") \+
we can evaluate and manipulate f in the usual way.  Examples:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(2,1);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "f(a,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"f(t,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 6 "Plot3d" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
200 "The easiest way to graph a function of two variables is using the
 plot3d command.  It will result in a surface in three dimensions.  Yo
u need to state the function and the ranges of the two variables." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot3d(f,-3..3,-3..3);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 142 "We could also graph f as an expression. \+
 Notice the slight difference in the plotting statement below (of cour
se, the graphs remain the same)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"plot3d(f(x,y),x=-3..3,y=-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
207 "Let's add some bells and whistles.  Axes=normal adds the coordina
te axes, we change the style, we select 3-d viewing region (this lis l
ike setting the window on a graphing calculator) and we give it a titl
e." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot3d(f,-4..4,-4..4, axes=no
rmal, style=surface, view=[-4..4,-4..4,-4..4],title=' wow ');" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Suppose we want to also show the p
lane " }{XPPEDIT 18 0 "z = x-y;" "6#/%\"zG,&%\"xG\"\"\"%\"yG!\"\"" }
{TEXT -1 43 "   in the same figure.  First we define it," }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 18 "p := (x,y) -> x-y;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "and now we
 are ready to plot it along with the original surface." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 92 "plot3d([f,p],-4..4,-4..4, axes=normal, style=[
wireframe,surface], view=[-4..4,-4..4,-4..4]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 231 "There is a second option to show two graphs in the \+
same figure.  First you define the graphs one at a time (use : rather \+
than ; ), and then you display them simultaneously.  This methods requ
ires that you activate the plot package." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 80 "G1 := plot3d(f,-3..3,-3..3,axes=normal):\nG2 := plot3d(p,-3..3,-
3..3,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "displa
y(G1,G2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 13 "Contour Plots" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 266 "Level curves can be generated with the contourplot comma
nd, which is also part of the plots package.  We need to communicate t
he function and the region in which we want to see the level curves.  \+
To view the level curves of our function for -4 < x , y < 4,  we enter
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "contourplot(f,-4..4,-4..4);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The same plot with colors (contou
rs = 8 controls the number of level curves)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "contourplot(f,-4..4,-4..4,contours=8,coloring=[red,bl
ue],filled=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We can also \+
control the values in f(x,y) = k by setting specific values of k.  For
 example:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "contourplot(f,-4..4,-4
..4,contours=[-1.5,-1,-0.5,0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Implicit Pl
ots" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Let's now turn to functions
 of three variables, and use " }{XPPEDIT 18 0 "g(x,y,z) = x^2+4*y^2-z^
2;" "6#/-%\"gG6%%\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*&\"\"%F-*$F(F,F-F-*$F
)F,!\"\"" }{TEXT -1 60 "   as an example.  First we define g in the fu
nction syntax." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g := x -> x^2+4*y
^2-z^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "Level surface require
 to use three dimensional implicit plots.  We need to communicate the \+
equation which is to be plotted and the ranges for the variables.  For
 example, the level surface for g = 0 becomes" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "implicitplot3d( g(x,y,z)=0, x=-5..5,y=-5..5, z=-5..5,
axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "By adding more p
oints, the surface looks a little bit nicer at the origin." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 82 "implicitplot3d( g(x,y,z)=0, x=-5..5,y=-5..5
, z=-5..5,axes=normal,numpoints=50000);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 262 "Here we display three level surfaces simultaneously. The
 figure shows the transition from a hyperboloid of two sheets into a h
yperboloid of one sheet by way of a cone.  It is sliced open, and you \+
need to move the figure around to get a good view of the surfaces." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "implicitplot3d( [g(x,y,z)=0, g(x,y
,z)=1, g(x,y,z)=-1], x=0..2,y=-2..2, z=-2..2,axes=normal,numpoints=500
00,style=wireframe,color=[red,blue,green],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Same thing, but with animation." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "animate( implicitplot3d, [g(x,y,z
) = c, x=-2..2,y=-2..2,z=-2..2,numpoints=5000],c=-1..1, frames=51);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Partial \+
Derivatives" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We have not restart
ed, and the definitions of f and g are still active." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "f(x,y);\ng(x,y,z);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 295 "In taking partial derivatives we have the same distincti
on between D and diff, as we had for functions of one variable.  D app
lies to functions, diff applies to expressions!  We illustrate this by
 showing the two options side by side.  The partial derivative of f wi
th respect to x is found by  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "D[
1](f)(x,y);\ndiff(f(x,y),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "O
f course the results are identical.  The partial with respect to y is \+
found from " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "D[2](f)(x,y);\ndiff(
f(x,y),y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "For higher order de
rivatives we process as follows" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
D[1,1](f)(x,y);\ndiff(f(x,y),x,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "Here is the mixed partial derivative  " }{XPPEDIT 18 0 "f[xxxy]
(x,y);" "6#-&%\"fG6#%%xxxyG6$%\"xG%\"yG" }{TEXT -1 12 "   both ways" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "D[1,1,1,2](f)(x,y);\ndiff(f(x,y),x
,x,x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Evaluation of a parti
al derivative is straightforward in the function syntax" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "D[1](f)(3,2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 113 "With diff this is a little bit more tedious.  We need tw
o steps: first differentiate, then substitute the values:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "diff(f(x,y),x);\nsubs(\{x=3, y=2\},%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "One advantage of diff is that you
 don't need to define the function in advance.  You can just enter the
 expression as is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diff( x^2 - 2
*x*y + exp(x*y) , x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Implicit Differentiation" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Implicit differentiation is also \+
possible.  We need to specify the equation, the dependent and the inde
pendent variable (in that order).  Here is an example" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "SFC := z^2 = x^2 - 4*y^2 +4*x*y*z;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "In case you wonder about the shape of thi
s surface:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "implicitplot3d(SFC,x=
-4..4,y=-4..4,z=-4..4,axes=normal, numpoints=5000);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 37 "The partial of z with respect to x is" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "implicitdiff(SFC,z,x);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 44 "Differentiation of z withrespect to y yields" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "implicitdiff(SFC,z,y);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 62 "Double check the long way.  First solve t
he equation SFC for z" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve( SFC
,z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Take the first answer and
 call this function h(x,y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "h := \+
(x,y) -> 2*x*y+(4*x^2*y^2+x^2-4*y^2)^(1/2);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 63 "Now take the derivateive of h with respect to x  and ca
ll it HX" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "HX := diff(h(x,y),x);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The result from implicit differ
entiation was (call it ZX)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ZX :=
 implicitdiff(SFC,z,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Now su
bstitute h(x,y) for z" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs( z = \+
h(x,y), ZX);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Is this equal to \+
HX" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Does this equal HX?" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "% - HX;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 
"BINGO!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 43 }
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