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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.

61. u = cos(x2y)

To determine

To verify: The conclusion of Clairaut’s Theorem that is uxy=uyx for the function u=cos(x2y) .

Explanation

The given function is, u=cos(x2y) .

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

ux=sin(x2y)(2xy)

ux=2xysin(x2y) (1)

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

2uxy=2x[sin(x2y)(1)+ycos(x2y)(x2)]=2xsin(x2y)2x3ycos(x2y)=2x[sin(x2y)+x2ycos(x2y)]

Hence, uxy=2x[sin(x2y)+x2ycos(x2y)] .

Differentiate the given function with respect to y and obtain uy .

uy=sin(x2y)(x2)

uy=x2sin(x2y) (2)

Differentiate equation (2) with respect to x and obtain the partial derivative, uyx

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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