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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 8 "Math 252" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT 257 53 "The Gradient and Tangent Planes for Le
vel Surfaces of" }{TEXT 259 1 " " }{TEXT 258 12 "G(x,y,z) = c" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{TEXT 260 1 "D" }
{TEXT -1 12 " in space ( " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }
{TEXT -1 45 " ) is the domain of the real-valued function " }{TEXT 
261 1 "G" }{TEXT -1 16 " .  That is, G: " }{XPPEDIT 18 0 "D->R^3" "6#f
*6#%\"DG7\"6$%)operatorG%&arrowG6\"*$%\"RG\"\"$F*F*F*" }{TEXT -1 8 " .
 If   " }{XPPEDIT 18 0 "L[c]" "6#&%\"LG6#%\"cG" }{TEXT -1 20 "  is the
 subset of  " }{TEXT 262 1 "D" }{TEXT -1 12 "  for which " }{TEXT 263 
12 "G(x,y,z) = c" }{TEXT -1 9 " , then  " }{XPPEDIT 18 0 "L[c]" "6#&%
\"LG6#%\"cG" }{TEXT -1 31 "  is a level surface of or for " }{TEXT 
264 1 "G" }{TEXT -1 19 "  for the constant " }{TEXT 265 1 "c" }{TEXT 
-1 21 " .  For example, if  " }{XPPEDIT 18 0 "G(x,y,z)=x^2+y^2+z^2" "6
#/-%\"GG6%%\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*$F(F,F-*$F)F,F-" }{TEXT -1 
7 "  and  " }{XPPEDIT 18 0 "c=9" "6#/%\"cG\"\"*" }{TEXT -1 9 " , then \+
 " }{XPPEDIT 18 0 "L[9]" "6#&%\"LG6#\"\"*" }{TEXT -1 74 "  is the sphe
re of radius 3 centered at the origin.  Different values of  " }{TEXT 
266 1 "c" }{TEXT -1 49 "  produce different spheres as level surfaces \+
of " }{TEXT 267 1 "G" }{TEXT -1 32 " .  Thus, for a given function  " 
}{TEXT 268 1 "G" }{TEXT -1 13 "  and value  " }{XPPEDIT 18 0 "c" "6#%
\"cG" }{TEXT -1 28 "   we may define a surface  " }{XPPEDIT 18 0 "Sigm
a" "6#%&SigmaG" }{TEXT -1 6 "  as  " }{XPPEDIT 18 0 "L[c]" "6#&%\"LG6#
%\"cG" }{TEXT -1 23 "  .  Now suppose that  " }{XPPEDIT 269 0 "alpha" 
"6#%&alphaG" }{TEXT -1 24 "(t) is a space curve in " }{TEXT 270 1 "S" 
}{TEXT -1 14 " .  That is,  " }{XPPEDIT 271 0 "alpha" "6#%&alphaG" }
{TEXT -1 2 ": " }{XPPEDIT 18 0 "[a,b]->Sigma" "6#f*6#7$%\"aG%\"bG7\"6$
%)operatorG%&arrowG6\"%&SigmaGF,F,F," }{TEXT -1 14 " .  Suppose   " }
{XPPEDIT 272 0 "alpha" "6#%&alphaG" }{TEXT -1 6 "(t) = " }{TEXT 273 
17 "<f (t),g(t),h(t)>" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "By the nature of our assu
mptions, the composition " }{TEXT 274 1 "G" }{TEXT -1 1 "(" }{XPPEDIT 
275 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 276 1 "t" }{TEXT -1 
16 ")) = c for each " }{TEXT 277 1 "t" }{TEXT -1 120 " .  The overall \+
effect of our composition function is that the function is a constant.
  It only assumes one value, c .  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 24 "Thus the derivative of  " }{TEXT 314 1 "G" }{TEXT -1 1 "(" }
{XPPEDIT 257 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 315 1 "t" }
{TEXT -1 20 ")) with respect to  " }{TEXT 316 1 "t" }{TEXT -1 44 "  is
  0 .  The chain rule thus tells us that" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 6 "      " }{XPPEDIT 18 0 "G[x]" "6#&%\"GG6#%\"xG" }{TEXT -1 
1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 317 
1 "t" }{TEXT -1 3 ")) " }{TEXT 319 3 "f '" }{TEXT -1 1 "(" }{TEXT 318 
1 "t" }{TEXT -1 7 ")  +   " }{XPPEDIT 18 0 "G[y];" "6#&%\"GG6#%\"yG" }
{TEXT -1 1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }
{TEXT 320 1 "t" }{TEXT -1 3 ")) " }{TEXT 322 2 "g'" }{TEXT -1 1 "(" }
{TEXT 321 1 "t" }{TEXT -1 6 ")  +  " }{XPPEDIT 18 0 "G[z];" "6#&%\"GG6
#%\"zG" }{TEXT -1 1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT 
-1 1 "(" }{TEXT 323 1 "t" }{TEXT -1 3 ")) " }{TEXT 325 2 "h'" }{TEXT 
-1 1 "(" }{TEXT 324 1 "t" }{TEXT -1 13 ")   =  0     " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "But this can be rewritten using gradient \+
and dot product as" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 5 "     " }
{TEXT 327 4 "grad" }{TEXT -1 3 " G(" }{XPPEDIT 256 0 "alpha" "6#%&alph
aG" }{TEXT -1 1 "(" }{TEXT 326 1 "t" }{TEXT -1 6 ")) \267  " }
{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 2 "'(" }{TEXT 328 1 "t" 
}{TEXT -1 10 ")  =  0   " }}{PARA 259 "" 0 "" {TEXT -1 4 "    " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 329 7 "Example" }{TEXT -1 306 ": Page 710
 number 19. We have the surface of the cone, z^2 = x^2+ y^2, and we ha
ve a space curve C: r(t)=(t*cos(t),t*sin(t),t) lies on the cone (can y
ou verify this?). Find the tangent plane which passes through the poin
t (0,Pi/2,Pi/2). Use Maple to display the surface, space curve, and th
e tangent plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(li
nalg): with(plots): with(student):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "x0:=0: y0:=Pi/2: z0:=Pi/2: X0:=vector([x0,y0,z0]);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "G:=(x,y,z)->x^2+y^2-z^2;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "K:=G(x0,y0,z0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gradG:=grad(G(x,y,z),[x,y,z]
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "N:=subs(x=x0,y=y0,z=z
0,op(gradG));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=vector(
[x,y,z]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tplane1:=evalm
(innerprod(N,X)=innerprod(N,X0));\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "C:=spacecurve([t*cos(t),t*sin(t),t],t=0..4*Pi,thickne
ss=2,axes=FRAME):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Obviously, t
plane1 is an equation for the tangent plane at " }{XPPEDIT 18 0 "X[0]
" "6#&%\"XG6#\"\"!" }{TEXT -1 3 "  ." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "zee:=solve(tplane1,z);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 58 "Now let\222s set up the plots, including labels for the a
xes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "tplane:=plot3d(zee,
x=-5..5,y=-5..5,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "graphG:=implicitplot3d(G(x,y,z)=K,x=-5..5,y=-5..5,z=-10..10,color=
cyan,numpoints=10000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Nvec:=spacecurve(evalm(X0
+t*N),t=0..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "display(C,tplane,graph
G,Nvec,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 8 "Pr
oblem:" }{TEXT -1 9 "  Given: " }{XPPEDIT 18 0 "G(x,y,z)=x^2+y+z^2-3" 
"6#/-%\"GG6%%\"xG%\"yG%\"zG,**$F'\"\"#\"\"\"F(F-*$F)F,F-\"\"$!\"\"" }
{TEXT -1 9 "    and  " }{XPPEDIT 18 0 "X[0]=(1,1,1)" "6#/&%\"XG6#\"\"!
6%\"\"\"F)F)" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "
(a) Use Maple to find the gradient of " }{TEXT 308 3 " G " }{TEXT -1 
5 " at  " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 65 "  and \+
an equation for the tangent plane to the level surface of  " }{TEXT 
309 1 "G" }{TEXT -1 17 "  that contains  " }{XPPEDIT 18 0 "X[0]" "6#&%
\"XG6#\"\"!" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "(
b) Plot the level surface of " }{TEXT 310 1 "G" }{TEXT -1 24 " , the t
angent plane at " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 
47 " , and a line that represents the gradient of  " }{TEXT 311 1 "G" 
}{TEXT -1 6 "  at  " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT 
-1 29 "  .  Include coordinate axes." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "with(linalg): with(plots): with(student):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x0:=1: y0:=1: z0:=1: X0:=vector([x0
,y0,z0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "G:=(x,y,z)->x^
2+y+z^2-3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "K:=G(x0,y0,z0
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gradG:=grad(G(x,y,z),
[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "N:=subs(x=x0,y
=y0,z=z0,op(gradG));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=
vector([x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tplane1
:=evalm(innerprod(N,X)=innerprod(N,X0));\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 59 "Obviously, tplane1 is an equation for the tangent plane a
t " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 3 "  ." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "zee:=solve(tplane1,z);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now let\222s set up the plots, inc
luding labels for the axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "tplane:=plot3d(zee,x=0..3,y=0..3,color=red):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 83 "graphG:=implicitplot3d(G(x,y,z)=K,x=0..3,y=0.
.3,z=0..3,color=cyan,numpoints=10000):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Nvec:=s
pacecurve(evalm(X0+t*N),t=0..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "disp
lay(tplane,graphG,Nvec,scaling=constrained,axes=BOXED);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "49" 0 }{VIEWOPTS 1 1 0 
3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }