denote two points in the simply connected region Rof the vector field F . The work
done can be expressed by performing a line integral of the equation (9.40) to obtain
W=
∫P 2 (x 2 ,y 2 ,z 2 )
P 1 (x 1 ,y 1 ,z 1 )
F·dr =
∫P 2
P 1
∇φ·dr =
∫P 2
P 1
dφ =φ|PP^21
or W=
∫P 2 (x 2 ,y 2 ,z 2 )
P 1 (x 1 ,y 1 ,z 1 )
F·dr =φ(x 2 , y 2 , z 2 )−φ(x 1 , y 1 , z 1 )
(9 .41)
which implies that the work done depends only on the end points P 1 and P 2 and is
thus independent of the path which joins these two points. Thus statement 2 above
implies statement 4. Note that this result does not necessarily hold for multiply
connected regions.
The line integral given by equation (9.41) being independent of the path of
integration which joins P 1 and P 2 can be expressed as
∫
C 1
F·dr =
∫
C 2
F·dr, (9 .42)
where the integral on the left is along a path C 1 and the integral on the right is along
a path C 2 ,where both paths go from P 1 to P 2 as illustrated in figure 9-2.
Figure 9-2. Paths of integration.
The integral (9.42) can be expressed in the form
∫
C
©F·dr = 0, (9 .43)
where the closed path Cgoes from P 1 to P 2 along the path C 1 and then from P 2 to P 1
along the path C 2 .The curves C 1 and C 2 are arbitrary so that the work done in going
