Calculus: Analytic Geometry and Calculus, with Vectors

11.2 Increments, chain rule, and gradients 571

at the point (xo,yo,zo), and find the equation of the plane tangent to the latter
surface at the latter point. /Ins.:

VF = yzi + azj + 1 vk

x - x_Y - Yo z- zo

Yozo

• xozo xoYo
  YOZO(X - xo) + xozo(y - Yo) + xoYo(z - zo) = 0.

(^15) Prove that if the plane 7r is tangent to the graph of the equationxyz = a',
then Tr and the three coordinate planes are the boundaries ofa tetrahedron
having volume 9a'/2.
(^16) Prove that if u, uzj ity, and u= are continuous over a spherical ball having
its center at Po(xo,yo,zo), then u cannot have evena local minimum or a local
maximum at PO unless
u=(xo,Yo,zo) = uy(xo,yo,zo) = u=(xo,Y0.zo)= 0.
Hint: Tell what can be done when the gradient at Po is not 0. If necessary, look
at (11.262).
17 Using the result of Problem 16, show that if
u x+Y+z
+x2+Y2+z2
attains a maximum at P(xo,yo,zo), then xo= yo = zo. With this assistance, find
the place where u is maximum and show that the maximum value of u is //2.
18 Remark: This is a remark for those who wish to see that gradients can be
used to introduce a fundamental idea that is often used to solve more difficult
problems. Suppose we are required to find numbers
x, y, z for which f (x,y,z) is a minimum or maximum when
x, y, z are required to satisfy a supplementary condition
of the form u(x,y,z) = c. To catch the idea, we suppose
that
f(x,Y,z) = x2 + Y2 + z2
and that the graph of u = c is a surface more or less like
that shown in Figure 11.291. It is easy to guess that if
a maximum or a minimum occurs at P(x,y,z), then
the gradients of f and u at P must have the same or
Figure 11.291
opposite directions and hence that there must be a constant X such that

Vf+AVu=0

or

(1) 0(f+Au) =0.-

We now make a profound observation. Finding values of x, y, z, and A for which

(1) holds is equivalent to finding values of x, y, z, and A for which the first partial

derivatives of the function w define by

(2) w(x,Y,z) = f(x,Y,z) + A(u(x,Y,z) - cl

are 0. All this leads us to an idea that can sound very strange and which might

be quite useless if it were not for the fact that it is the very useful idea that the