   Chapter 15.1, Problem 52E

Chapter
Section
Textbook Problem

(a) In what way are the theorems of Fubini and Clairaut similar?(b) If f(x, y) is continuous on [a, b] × [c, d] and g ( x , y ) = ∫ a x ∫ c y f ( s , t ) d t   d s for a < x < b, c < y < d, show that gxy = gyx = f(x, y).

(a)

To determine

To explain: The similarity of the Fubini and Clairaut theorems.

Explanation

Theorems used:

Clairaut’s theorem:

Suppose f is a real-valued function of two variables x,y and f(x,y) is defined on an open subset U of 2 . Suppose further that both the second-order mixed partial derivatives fxy(x,y) and fyx(x,y) exist and are continuous on U. Then, fxy=fyx on all of U.

Fubini’s theorem:

If f is continuous on the rectangular region, R={(x,y)|axb,cyd} , then Rf(x,y)dA=

(b)

To determine

To show: gxy=gyx=f(x,y)

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