   Chapter 16.8, Problem 8E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫c F · dr. In each case C is oriented counterclockwise as viewed from above.8. F(x, y, z) = i + (x + yz) j + (xy - √z) k, C is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant

To determine

To evaluate: The value of CFdr by the use of Stokes’ theorem.

Explanation

Given data:

The field is F(x,y,z)=i+(x+yz)j+(xyz)k and C is boundary of part of plane 3x+2y+z=1 in first octant.

Formula Used:

Write the expression for curl of F(x,y,z)=Pi+Qj+Rk .

curlF=|ijkxyzPQR|

curlF=(RyQz)i(RxPz)j+(QxPy)k (1)

Write the expression for the Stokes’ theorem.

CFdr=ScurlFdS (2)

Here,

S is surface.

Consider surface S, z=g(x,y) is in upward orientation. Write the expression for surface integral of F over surface S.

ScurlFdS=D(PgxQgy+R)dA (3)

Here,

A is area.

Find the value of curlF by using equation (1).

curlF=((xyz)y(x+yz)z)i((xyz)x(1)z)j+((x+yz)x(1)y)k=((x(1)0)(0+y(1)))i((y(1)0)0)j+((1+0)0)k=(xy)iyj+k

The surface S is a planar region which is enclosed by plane C. The expression for surface S, D={(x,y)|0x13,0y12(13x)} is,

3x+2y+z=1z=13x2y

Hence the equation is in the form of z=g(x,y) .

The curve C is oriented in counter-clockwise, so the orient the surface S upward.

Compare the equations curlF=Pi+Qj+Rk and curlF=(xy)iyj+k .

P=xyQ=yR=1

Find the value of ScurlFdS by the use of equation (3)

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

Find the mean for the scores in the following frequency distribution table: X f 6 1 5 4 4 2 3 2 2 1

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Evaluate the integral. 01(1+r)3dr

Calculus (MindTap Course List)

In problems 23-58, perform the indicated operations and simplify. 47.

Mathematical Applications for the Management, Life, and Social Sciences

Solve each equation and check: 82y=4

Elementary Technical Mathematics

Explain how you would find the inverse of a nonsingular matrix.

Finite Mathematics for the Managerial, Life, and Social Sciences

True or False: Let f(x) = x2. The Intermediate Value Theorem guarantees there is for which f(w) = 0.

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

True or False: The function f is continuous at (0, 0), where

Study Guide for Stewart's Multivariable Calculus, 8th 