I like to think in higher dimensions. If you follow a flow through a potential field, then the gain of energy is determined by your path, and is zero if it is a closed curve. What happens inside is determined by what's happening at the boundary. You can write this as [imath]\int_a^b +\int_b^a=0. [/imath] For a concrete example, you may consider climbing stairs between two floors: the potential is the gravity, the energy gain the difference in potential energy. Your energy gain is [imath] F(3rd)-F(2nd)[/imath] no matter what you did in between, climbing back and forth, pausing, or taking two steps at a time.
I am sure you have seen a lot of examples and maybe a few proofs, too, so it's hard to imagine what else can be done. This is how I see it: it is technically correct, so I accept it. A flow through a vector field is my intuition. If you break this picture down to two dimensions, the graph of a function, you end up with Riemann sums.
