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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Determine whether each of the following functions is a solution of Laplace’s equation uxx + uyy = 0.

(a) u = x2 + y2

(b) u = x2y2

(c) u = x2 + 3xy2

(d) u = ln x 2 + y 2

(e) u = sin x cosh y + cos x sinh y

(f) u = ex cos yey cos x

(a)

To determine

Whether the function u=x2+y2 is a solution of Laplace’s equation, uxx+uyy=0 .

Explanation

The given function is, u=x2+y2 .

Take the partial derivative of the given function with respect to x and obtain ux .

ux=x(x2+y2)=x(x2)+x(y2)=2x+0=2x

Thus, ux=2x . (1)

Take the partial derivative the equation (1) with respect to x and obtain uxx .

2ux2=x(2x)=2x(x)=2(1)=2

Thus, the partial derivative, 2ux2=2 .

Take the partial derivative of the given function with respect to y and obtain uy

(b)

To determine

Whether the function u=x2y2 is a solution of Laplace’s equation, uxx+uyy=0 .

(c)

To determine

Whether the function u=x3+3xy2 is a solution of Laplace’s equation, uxx+uyy=0 .

(d)

To determine

Whether the function u=lnx2+y2 is a solution of Laplace’s equation, uxx+uyy=0 .

(e)

To determine

Whether the function u=sinxcoshy+cosxsinhy is a solution of Laplace’s equation, uxx+uyy=0 .

(f)

To determine

Whether the function u=excosyeycosx is a solution of Laplace’s equation, uxx+uyy=0 .

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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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