# How to prove that work due to conservative forces are independent of path?

The Wikipedia article on conservative forces says,

A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions: [...] 3. The force can be written as the negative gradient of a potential, $$\Phi$$: $$\vec F = - \vec \nabla \Phi$$.

Source: https://en.wikipedia.org/wiki/Conservative_force

I don't understand why $$\Phi$$ having spatial derivatives equal to $$\vec F$$'s components implies that the work done by the force $$W = -\Delta \Phi$$ is independent of the path a test particle takes.

• Does not the next section of the Wikipedia article answer your question? Sep 17, 2022 at 4:56
• perhaps look at the fundamental theorem of calculus for line integrals which tells you that the line integral of the gradient of a scalar depends only on the end points. It's not really anything about having "spatial derivatives" it's more about how F be written in terms of what derivatives. as curl, gradient, divergence, etc. are all different ways of differentiating--each with different properties. Sep 17, 2022 at 6:42