   Chapter 16.8, Problem 7E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫c F · dr. In each case C is oriented counterclockwise as viewed from above.7. F(x, y, z) = (x + y2) i + (y + z2) j + (z + x2) k, C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)

To determine

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

Explanation

Given data:

The field is F(x,y,z)=(x+y2)i+(y+z2)j+(z+x2)k and C is triangle with vertices (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.

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

curlF=[((z+x2)y(y+z2)z)i((z+x2)x(x+y2)z)j+((y+z2)x(x+y2)y)k]=(0(0+2z))i((0+2x)0)j+(0(0+2y))k=2zi2xj2yk

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.

Compare the equations curlF=Pi+Qj+Rk and curlF=2zi2xj2yk

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