   Chapter 16, Problem 33RE

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫C F · dr, where F(x, y, z) = xy i + yz j + zx k, and C is the triangle with vertices (1, 0. 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.

To determine

To evaluate: The value of CFdr , where F(x,y,z)=xyi+yzj+zxk using Stokes’ Theorem.

Explanation

Given data:

The field is F(x,y,z)=xyi+yzj+zxk and C is triangle with vertices are (1,0,0) , (0,1,0) and (0,0,1) .

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.

Consider the general expression for field as follows.

F(x,y,z)=Pi+Qj+Rk (4)

Consider the field expression as follows.

F(x,y,z)=xyi+yzj+zxk (5)

Compare equations (4) and (5).

P=xyQ=yzR=zx

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)i(z0)j+(0x)k=yizjxk

Consider the surface S is a planar region which is enclosed by curve C. The expression for surface S, D={(x,y)|0x1,0y1x} since it is a plane is,

x+y+z=1z=1xy

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

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