   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

Solve the equations in Exercises 126. 6x(x2+1)2(x2+2)48x(x2+1)3(x2+2)3(x2+2)8=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Factor completely: 8m210m3

Elementary Technical Mathematics

For , f′(x) =

Study Guide for Stewart's Multivariable Calculus, 8th 