I was reading Goldstein's book on mechanics and came across this theorem:
$F(r) = - \nabla V(r)$ is a necessary and sufficient condition of the force field being conservative.
So far, I have understood the condition of a force being conservative as path independence: $\int_{\text{closed loop}} F \cdot ds = 0$.
The new condition was justified by a brief argument which I don't follow:
The existence of V can be inferred intuitively by a simple argument. If $W_{12}$ is independent of the path of integration between the end points 1 and 2, it should be possible to express $W_{12}$ as the change in quantity that depends only upon the positions of the end points.
I follow that for the work done must depend only upon the end points $W = W(start, end)$ and to ensure the path integral is always zero the "return trip" must cancel out the "outgoing trip", i.e., $W = W(start, end) = -W(end, start)$. But how do I go from this to the form given above?