   Chapter 16.8, Problem 9E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫c F · dr. In each Case C is oriented counterclockwise as viewed from above.9. F(x, y, z) = xy i + yz j + zx k, C is the boundary of the part of the paraboloid z = 1 - x2 - y2 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)=xyi+yzj+zxk and C is boundary of part of paraboloid z=1x2y2 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 the 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=((zx)y(yz)z)i((zx)x(xy)z)j+((yz)x(xy)y)k=(0y(1))i(z(1)0)j+(0x(1))k=yizjxk

The surface S is a paraboloid region which is enclosed by plane C. The expression for surface S, D={(r,θ)|0x1,0yπ2} in polar coordinate system is,

z=1x2y2

Here,

x=rcosθandy=rsinθ

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=yizjxk .

P=yQ=zR=x

Find the value of ScurlFdS by using equation (3).

ScurlFdS=D((y)(1x2y2)x(z)(1x2y2)y+(x))dA=D(y(2x)+z(2y)x)dA=D(2xy2yzx)dA=D(2xy2y(1x2y2)x)dA{z=1x2y2

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 more solutions based on key concepts 