   Chapter 16.8, Problem 17E

Chapter
Section
Textbook Problem

A particle moves along line segments from the origin to the points (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the origin under the influence of the force field F(x, y, z) = z2 i + 2xy j + 4y2 k Find the work done.

To determine

To find: The work done by particle under influence of the force field F(x,y,z)=z2i+2xyj+4y2k.

Explanation

Given data:

The force field is F(x,y,z)=z2i+2xyj+4y2k.

The particle moves along line segments from the origin to the points (1,0,0),(1,2,1),(0,2,1) and back to origin.

Formula Used:

The work done by force F(x,y)=P(x,y)i+Q(x,y)j moving over curve C is,

W=CFdr (1)

Where,h is work done, and P and Q have continuous partial derivatives.

The curl of F(x,y,z)=Pi+Qj+Rk is given by,

curlF=|ijkxyzPQR|

curlF=(RyQz)i(RxPz)j+(QxPy)k (2)

The Stokes’ theorem is,

CFdr=ScurlFdS (3)

The surface integral of F over surface S is given by,

ScurlFdS=D(PgxQgy+R)dA (4)

Calculation:

Consider the surface S as a region of plane that is enclosed path by the particle and hence, the surface is the portion of the plane,

z=12yfor0x1,0y2

Hence the equation is in the form of z=g(x,y) and the orientation of surface S is in upward direction.

Find the value of curlF by using equation (2).

curlF=((4y2)y(2xy)z)i((4y2)x(z2)z)j+((2xy)x(z2)y)k=(4(2y)0)i(02z)j+(2y(1)0)k{t(k)=0,t(t)=1,t(t2)=2t}=8yi+2zj+2yk

Compare the equations curlF=Pi+Qj+Rk and curlF=8yi+2zj+2yk

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