Page 401 - Advanced engineering mathematics

12.4 Independence of Path and Potential Theory 381

In particular, if C is a closed path in D, then

F · dR = 0.
C

The last conclusion follows from the fact that, if the path is closed, then the initial point P 0
and the terminal point P 1 are the same and equation (12.4) yields 0 for the value of the integral.

Another consequence of F having a potential function is independence of path. We say that

F · dR is independent of path in D if the value of this line integral for any path C in D
C
depends only on the endpoints of C. Another way of putting this is that

FdR = F · dR
C 1 C 2
for any paths C 1 and C 2 in D having the same initial point and the same terminal point
in D. In this case, the route is unimportant - the only thing that matters is where we
start and where we end. By equation (12.4), existence of a potential function implies
independence of path.

THEOREM 12.3

If F is conservative in D, then FdR is independent of path in D.
˙
C
Proof Let ϕ be a potential function for F in D.If C 1 and C 2 are paths in D having initial point
P 0 and terminal point P 1 , then

F · dR = ϕ(P 1 ) − ϕ(P 0 ) = F · dR.
C 1 C 2
Independence of path is equivalent to the vanishing of integrals around closed paths.

THEOREM 12.4

F · dR is independent of path in D if and only if
C

F · dR = 0
C
for every closed path in D.

Proof To go one way, suppose first that F · dR=0 for every closed path in D and let C 1 and
C
C 2 be paths in D from P 0 to P 1 . Form a closed path C by starting at P 0 , moving along C 1 to P 1 ,

and then reversing orientation to move along −C 2 from P 1 to P 0 . Then C = C 1 (−C 2 ) and

F · dR = 0 = F · dR − F · dR,
C C 1 C 2
implying that

F · dR = F · dR.
C 1 C 2

This makes F · dR independent of path in D.
C

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October 14, 2010 14:53 THM/NEIL Page-381 27410_12_ch12_p367-424