   Chapter 16.2, Problem 41E

Chapter
Section
Textbook Problem

Find the work done by the force fieldF(x, y, z) = ⟨x− y2, y − z2, z − x2⟩on a particle that moves along the line segment from (0, 0, 1) to (2, 1, 0).

To determine

To find: The work done by the force field F(x,y,z)=xy2,yz2,zx2 .

Explanation

Given data:

The force field is F(x,y,z)=xy2,yz2,zx2 .

The particle moves along the line segment from the point (0,0,1) to (2,1,0) .

Formula used:

Write the expression to find the work done by the force field F(x,y,z) along the line segment.

CFdr=abF(r(t))r(t)dt (1)

Here,

r(t) is the vector function of the line segment,

a is the lower limit of curve, and

b is the upper limit of the curve.

Write the expression to find the parametric equations for a line segment through the point (x0,y0,z0) and parallel to the direction vector v=a,b,c .

x=x0+at,y=y0+bt,z=z0+ct (2)

Write the expression to find the director vector v=a,b,c for a line segment from the point (x0,y0,z0) to (x1,y1,z1) .

a,b,c=x1x0,y1y0,z1z0 (3)

Calculation of direction vector v=a,b,c for a line segment:

Substitute 0 for x0 , 0 for y0 , 1 for z0 , 2 for x1 , 1 for y1 , and 0 for z1 in equation (3),

a,b,c=20,10,01=2,1,1

Calculation of parametric equations of line segment:

Substitute 0 for x0 , 0 for y0 , 1 for z0 , 2 for a , 1 for b , and (1) for c in equation (2),

x=0+(2)t,y=0+(1)t,z=1+(1)tx=2t,y=t,z=1t

Write the vector function from the parametric equations as follows.

r(t)=2t,t,1t

Write the force field as follows.

F(x,y,z)=xy2,yz2,zx2 (4)

Calculation of F(r(t)) :

Substitute 2t for x , t for y , 1t for z in equation (4),

F(r(t))=2tt2,t(1t)2,(1t)(2t)2=2tt2,t(1+t22t),(1t)4t2=2tt2,3t1t2,1t4t2

Calculation of r(t) :

To find the derivative of the vector function, differentiate each component of the vector function

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