Proof of Equation 8-2

Figure 8-4Z? shows an arbitrary round trip for a particle that is acted upon by a single force. The particle moves from an initial point a to point b along path 1, and then back to point a along path 2. The force does work on the particle as the particle moves along each path. Without worrying about where positive work is done and where negative work is done, let us just represent the work done from a to b along path 1 as and the work done from b back to a along path 2 as .If the force is conservative, then the net work done during the round trip must be zero:

and thus

In words, the work done along the outward path must be the negative of the work done along the path back.

Let us now consider the work done on the particle by the force when the particle moves from a to b along path 2, as indicated in Fig. 8-4a. If the force is conservative, that work is the negative of :

. Substituting for in Eq. 8-3, we obtain

which is what we set out to prove.