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

[Za Kh]

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]

Hi ZA,

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

 

[Za Kh]

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.