# Why is a conserved vector field a gradient of a certain func

I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.

BvU
Homework Helper
2019 Award
Hi ZA,

You of course googled conservative vector field but there must be something that isn't clear to you. What specifically ?

Hi ZA,

You of course googled conservative vector field but there must be something that isn't clear to you. What specifically ?
Hi! I've figured this out, by the stocks theorem and that the work done by a conservative force on a closed curve is zero but why it isn't true for the non conservative force?

HallsofIvy
That comes from the "Fundamental Theorem of Calculus", that if F is a differentiable function such that F'= f then $\int_a^b= f(b)- f(a)$. A "conservative force" is the derivative of some "energy function". The integral depends only on the end points, not the path between the end points. The integral around a closed path, since the "end points" are the same point, is 0.