Conservative vector field

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

<math>\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</math>

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.