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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.

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  • $\begingroup$ Does not the next section of the Wikipedia article answer your question? $\endgroup$
    – Farcher
    Commented Sep 17, 2022 at 4:56
  • $\begingroup$ 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. $\endgroup$ Commented Sep 17, 2022 at 6:42

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