   Chapter 14, Problem 6RCC

Chapter
Section
Textbook Problem

What does Clairaut’s Theorem say?

To determine

To state: The Clairaut’s Theorem.

Explanation

Clairaut’s Theorem:

“If f is a function define on a disk D and has second-ordered partial derivatives with continuous. Suppose the disk D contains the point (a,b) , then fxy=fyx ”.

Example:

Consider the function u=exysiny .

Differentiate the given function with respect to x and obtain ux .

ux=siny[exy(y)] .

ux=yexysiny (1)

Differentiate the equation (1) with respect to y and obtain the partial derivative, uxy .

2uxy=yexy(cosy)+siny[exy(1)+yexy(x)]=yexycosy+exysiny+xyexysiny=exy(ycosy+siny+xysiny)

Hence, uxy=exy(ycosy+siny+xysiny) .

Differentiate the given function with respect to y and obtain uy

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