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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

Assume that all the given functions are differentiable.

46. If u = f(x, y), where x = es cos t and y = es sin t, show that

( u x ) 2 + ( u y ) 2 = e 2 s [ ( u s ) 2 + ( u t ) 2 ]

To determine

To show: The equation (ux)2+(uy)2=e2s[(us)2+(ut)2] if u=f(x,y) , where x=escost and y=essint .

Explanation

The function is z=u(x,y) .

The partial derivative, us is computed as follows,

us=uxxs+uyys=uxs(escost)+uys(essint)=uxescost+uyessint

Thus, the partial derivative is us=uxescost+uyessint .

Obtain the partial derivative, ut .

ut=uxxt+uyyt=uxt(escost)+uyt(essint)=ux(essint)+uy(escost)

Thus, the partial derivative is ut=ux(essint)+uy(escost) .

Find the value of [us]2 .

[us]2=[uxescost+uyessint]2=[(ux)2(escost)2]+2[(ux)(escost)(uy)(essit)]+[(uy)2(essint)2]=(ux)2e2scos2t+2(ux)(uy)e2scostsint+(uy)2e2ssin2t=e2s[(ux)2cos2t+2(ux)(uy)costsint+(uy)2sin2t]

Thus, the value (us)2=e2s[(ux)2cos2t+2(ux)(uy)costsint+(uy)2sin2t] .

Find the value of [ut]2 .

[ut]2=[ux.(essint)+uy

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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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