   Chapter 16.8, Problem 2E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS.2. F(x, y, z) = x2sin z i + y2 j + xy k. S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-plane, oriented upward

To determine

To evaluate: The expression ScurlFdS by using Stokes’ theorem.

Explanation

Given data:

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

F(x,y,z)=x2sinzi+y2j+xyk (1)

And S is the part of the paraboloid that is,

z=1x2y2 (2)

Equation (1) lies above the xy-plane, oriented upward.

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (3)

The paraboloid in equation (2) intersects the xy-plane in the circle x2+y2=1,z=0 . This boundary curve C must be oriented in the counter-clockwise direction. Therefore, the vector equation of C is,

r(t)=costi+sintj,0t2π (4)

Differentiate equation (4) with respect to t,

r(t)=ddt(costi+sintj)=sinti+costj

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

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

F(r(t))=(cost)2(sin0)i+(sint)2j+(cost)(sint)k=sin2tj+sintcostk

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

ScurlFdS=CFdr=02πF(r(t))r(t)dt

Substitute sin2tj+sintcostk for F(r(t))

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