   Chapter 16.8, Problem 5E

Chapter
Section
Textbook Problem

F(x, y, z) = xyz i + xy j + x2yz k. S consists of the top and the four sides (but not the bottom) of the cube with vertices (±1, ±1, ± 1), oriented outward

To determine

To evaluate: The expression ScurlFdS by the use of Stokes’ theorem.

Explanation

Given data:

Consider the expression for the vector field F(x,y,z) ,

F(x,y,z)=xyzi+xyj+x2yzk (1)

And S is the cube with vertices (±1,±1,±1) and oriented outward.

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (2)

Consider C is the square in the plane z=1 .

Consider S1 is the original cube without bottom and S2 is bottom face of the cube. If S1 and S2 are oriented surfaces with same oriented boundary curve C and both surfaces satisfy the hypotheses of Stokes' Theorem. Therefore, consider the following expression.

S1curlFdS=CFdr=S2curlFdS

Calculate curlF by using equation (1),

curlF=|ijkddxddyddzxyzxyx2yz|=i(d(x2yz)dyd(xy)dz)j(d(x2yz)dxd(xyz)dz)+k(d(xy)dxd(xyz)dy)=x2zi+(xy2xyz)j+(yxz)k

For the surface S2 , choose n=k . So that C has the same orientation for both surfaces S1 and S2 .

Calculate curlFn

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