0
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ Does not the next section of the Wikipedia article answer your question? $\endgroup$
    – Farcher
    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. $\endgroup$
    – Silly Goose
    Sep 17, 2022 at 6:42

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.