Begin2.DVI

Note that each of the line integrals requires knowing the values of x,y and z

along a given curve Cand these values must be substituted into the integrand and

after this substitution the summation process reduces to an ordinary integration.

Work Done.

Consider a particle moving from a point POto a point P 1 along a curve Cwhich

lies in a force field F
=F
(x, y, z).At each point (x, y, z)on the curve there are force

vectors acting on the particle as illustrated in figure 6-19.

Figure 6-19. Moving along a curve Cin a force field F
.

Examine the particle at a general point (x, y, z)on the given curve C. Construct

the position vector
r, the force vector F ,
and the tangent vector d
r acting at this

general point on the curve. The line integral

WP 0 P 1 =

∫

C

F
·d
r =

∫P 1

P 0

F
·d
r

ds ds =

∫P 1

P 1

F
·ˆetds

is a summation of the tangential component of the force times distance traveled

along the curve C. Consequently, the above integral represents the work done in

moving through the force field from point P 0 to P 1 along the curve C.

Example 6-30. Let a particle with constant mass mmove along a curve C

which lies in a vector force field F
=F
(x, y, z ).Also, let
r denote the position vector

of the particle in the force field and on the curve C. As the particle moves along the

curve, at each point (x, y, z )of the curve, the particle experiences a force F
(x, y, z )