Conservative vector field

A conservative field or conservative vector field has a curl of zero:

$\nabla \times {\vec {V}}=(\ \ {\frac {\partial V_{z}}{\partial y}}-{\frac {\partial V_{y}}{\partial z}},\ \ \ \ {\frac {\partial V_{x}}{\partial z}}-{\frac {\partial V_{z}}{\partial x}},\ \ \ \ {\frac {\partial V_{y}}{\partial x}}-{\frac {\partial V_{x}}{\partial y}}\ \ )=0$

Its significance is that the line integral of a conservative field, such as a physical force, is independent of the path chosen. In physics, this means that the potential energy (which is determined by a conservative force field) of a particle at a given position is independent of how a particle was moved to its position.

The proof of this uses Stokes' Theorem. Since the curl is zero, any line integral around a closed loop is zero. If there are two paths from point A to point B, the first path from A to B, followed by the second path in reverse direction from B back to A, constitutes a closed loop, so its line integral is zero. But that's the sum of the first path integral and the negative of the second path integral, so the integrals are equal.