   Chapter 16.8, Problem 4E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS.4. F(x, y, z) = tan-1(x2yz2) i + x2y j + x2z2 k. S is the cone x   =   x 2 +   z 2 ,   0   ≤   x   ≤   2 , oriented in the direction of the positive x-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)=tan1(x2yz2)i+x2yj+x2z2k (1)

And S is the cone that is,

x=y2+z2,0x2 (2)

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (3)

Consider the boundary curve C is the circle y2+z2=4,x=2 . This boundary curve C must be oriented in the counter-clockwise direction. Therefore, the vector equation of C is,

r(t)=2i+2costj+2sintk,0t2π (4)

Differentiate equation (4) with respect to t.

r(t)=ddt(2i+2costj+2sintk)=2sintj+2costk

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

Substitute 2 for x, 2cost for y and 2sint for z in equation (1) to find F(r(t)) ,

F(r(t))=tan1(22(2cost)(2sint)2)i+22(2cost)j+22(2sint)2k=tan1(32costsin2t)i+8costj+16sin2tk

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

ScurlFdS=CFdr=02πF(r(t))r(t)dt

Substitute tan1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Evaluate the integral, if it exists. 01ex1+e2xdx

Single Variable Calculus: Early Transcendentals, Volume I

Change 65 cm to m.

Elementary Technical Mathematics

29. Write the first four terms and the 10th term of the sequence whose nth term is

Mathematical Applications for the Management, Life, and Social Sciences

The graph of x = cos t, y = sin2 t is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 