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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

Use the Chain Rule to find the indicated partial derivatives.

23. w = xy + yz + zx, x = r cosθ, y = r sinθ, z = ;

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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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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