   Chapter 14.5, Problem 46E

Chapter
Section
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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