   Chapter 16.8, Problem 10E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫c F · dr. In each Case C is oriented counterclockwise as viewed from above.10. F(x, y, z) = 2y i + xz j + (x + y) k, C is the curve of intersection of the plane z = y + 2 and the cylinder x2 + y2 = l

To determine

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

Explanation

Given data:

The field is F(x,y,z)=2yi+xzj+(x+y)k and C is curve of intersection of plane z=y+2 and cylinder x2+y2=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 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 the area.

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

curlF=((x+y)y(xz)z)i((x+y)x(2y)z)j+((xz)x(2y)y)k=((0+1)x(1))i((1+0)0)j+(z(1)2(1))k=(1x)ij+(z2)k

The surface S is a region which is enclosed by plane C. The expression for surface S, D={(r,θ)|0x1,0y2π} in polar coordinate system is,

z=y+2

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 orient the surface S in upwards.

Compare the equations curlF=Pi+Qj+Rk and curlF=(1x)ij+(z2)k .

P=1xQ=1R=z2

Find the value of ScurlFdS by using equation (3)

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