   Chapter 16.8, Problem 6E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS.6. F(x, y, z) = exy i + exz j + x2z k, S is the half of the ellipsoid 4x2 + y2 + 4z2 = 4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis

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)=exyi+exzj+x2zk (1)

And S is the half of the ellipsoid that is,

4x2+y2+4z2=4 (2)

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (3)

Consider the boundary curve C is the circle x2+z2=1,y=0 . This boundary curve C must be oriented in the counter-clockwise direction when viewed from the right. Therefore, the vector equation of C is,

r(t)=cos(t)i+sin(t)k=costisintk,0t2π

Differentiate the equation with respect to t.

r(t)=ddt(costisintk)=sinticostk

Find the expression for F(r(t)) .

Substitute cost for x, 0 for y and sint for z in equation (1) to find F(r(t)) ,

F(r(t))=e(cost)(0)i+e(cost)(sint)j+(cost)2(sint)k=0+ecostsintjcos2tsintk=ecostsintjcos2tsintk

Write the expression for the Stokes’ theorem in equation (3),

ScurlFdS=CFdr=02π

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