   Chapter 14.5, Problem 23E

Chapter
Section
Textbook Problem

Use the Chain Rule to find the indicated partial derivatives.23. w = xy + yz + zx, x = r cosθ, y = r sinθ, z = rθ; ∂ w ∂ r , ∂ w ∂ θ when r = 2, θ = π/2

To determine

To find: The value of wrandwθ using chain rule if w=xy+yz+zx,x=rcosθ,y=rsinθandz=rθ when r=2andθ=π2 .

Explanation

Given:

The function is, w=xy+yz+zx .

The value of r=2andθ=π2 .

Calculation:

Substitute r=2andθ=π2 in x ,

x=(2).cos(π2)=2.(0)(cos(π2)=0)=0

Thus, the value of x=0 .

Substitute r=2andθ=π2 in y ,

y=2.sin(π2)=2.(1)(sin(π2)=1)=2

Thus, the value of y=2 .

Substitute r=2andθ=π2 in z ,

z=2.(π2)=π

Thus, the value of z=π .

The partial derivative wr using chain rule is computed as follows,

wr=wxxr+wyyr+wzzr=x(xy+yz+zx)r(rcosθ)+y(xy+yz+zx)r(rsinθ)                                                     +z(xy+yz+zx)r(rθ)=(y+z)(cosθ)+(x+z)(sinθ)+(x+y)(θ)

Thus, the partial derivative wr=(y+z)(cosθ)+(x+z)(sinθ)+(y+x)(θ) .

Substitute the respective values in wr and obtain the required value.

wr=(y+z)(cosθ)+(x+z)(sinθ)+(y+x)(θ)=(2+π)

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