gradient theorem

The fundamental theorem of calculus says that integrating a derivative over an interval just gives the difference of the original function at the two endpoints — the messy accumulation collapses to two evaluations. The gradient theorem is exactly this statement lifted to curves in space: integrating the gradient of a function along a path gives the difference of the function at the path's two ends, no matter how the path wanders in between.

Precisely, if phi is a scalar field and C is a curve from point A to point B, then integral over C of grad phi dot dr = phi(B) - phi(A). The vector field grad phi is the multivariable derivative of phi; the line integral accumulates it along C; and the answer sees only the endpoints. This is the reason a conservative field's work integral is path-independent: once you know the field is grad phi, you never compute the integral at all — you just subtract two potential values. It is the cleanest of the integral theorems, with the lowest hypotheses.

The gradient theorem is the keystone connecting four ideas that turn out to be the same: a field being a gradient, having a potential, having path-independent line integrals, and having zero circulation around closed loops. In physics it is the statement that work done by a conservative force equals the drop in potential energy, the very definition of potential energy. It is also the genuine ancestor of the bigger theorems — Green's, Stokes', and the divergence theorem are all higher-dimensional fundamental theorems of calculus cut from the same cloth.