   Chapter 14.5, Problem 45E

Chapter
Section
Textbook Problem

Assume that all the given functions are differentiable.45. If z = f (x, y), where x = r cos θ and y = r sin θ, (a) find ∂z/∂r and ∂z/∂θ and (b) show that ( ∂ z ∂ x ) z + ( ∂ z ∂ y ) z = ( ∂ z ∂ r ) z + 1 r 2 ( ∂ z ∂ θ ) z

(a)

To determine

To find: The values of zrandzθ if z=f(x,y) , where x=rcosθ and y=rsinθ .

Explanation

The given function is, z=f(x,y) .

Take the partial derivative of x and y with respect to r ,

xr=r(rcosθ)=cosθyr=r(rsinθ)=sinθ

Thus, the partial derivates, xr=cosθandyr=sinθ .

Take the partial derivative of x and y with respect to θ ,

xθ=θ(rcosθ)=rsinθyθ=θ(rsinθ)=rcosθ

Thus, the partial derivates, xθ=rsinθandyθ=rcosθ .

The partial derivative, zr is computed as follows,

zr=zxxr+zyyr=zx

(b)

To determine

To show: The equation (zx)2+(zy)2=(zr)2+1r2(zθ)2 .

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