Calculus: Analytic Geometry and Calculus, with Vectors

562

12 The function u for which u(0,0) = 0 and

2xy

(1) u(x,Y) =(x2 +y2)2

can be considered more than once. Show that

Partial derivatives

(2) usz(x,y) = 24xy(x22+y2)4 uyy(x,y) = 24xv (z^2 +y2)4

and hence

a2u a2u

(3) axe + aye = u==(x,Y) + uvv(x,y) = 0

when x2 + y2 54 0.Show that (3) holds when x = y = 0. Show that

(4) lim u (x,z) = oo.

x-.O

13 Formulas more or less like

(1) F(x) = f bf(x,y) dy

often appear in pure and applied mathematics. We suppose that, for each x

in some interval, the integral in the right member of (1) exists and is a number

F(x). More advanced courses set forth conditions under which F'(x) exists and

can be obtained by "differentiating with respect to x under the integral sign"

so that

(2) F'(x)

=Jab [' fx(x,Y)] dy.

When this procedure is correct, we can combine (1) and (2) to obtain the formula

(3) dxfab f(x,y) dy = fabL' f(x,y)] dy.

Verify that (3) is correct when

(a) a = 0, b = 1, f(x,y) = x + y

(b) a = 0, b = 1, f(x,y) = x2 + y2

(c) a = 1, b = 2, f(x,y) =

sin xy

x

11.2 Increments, chain rule, and gradients

tion, we suppose that u is a function of three variablesx, y, z and we restrict

attention to a region R in E3 over which u is continuous and hascon-

tinuous partial derivatives u, u,,, anduZ. In many examples the region

R is the whole E3, but this is not necessarilyso. Whenever a useful

purpose is served, we can regard u(x,y,z) as being the temperature or

pressure or potential or density or humidity at the point P(x,y,z). While

u(x,y,z) cannot be a vector, it can be the scalarcomponent in some par-