I get that a force dependent on path is non-conservative or a force which does non-zero work in completing a cycle. But how do you prove that mathematically.
Let's assume I have a force F= 2yi + x^2j. Now how do I conclude that it's work done is path dependent?
My attempt Work done=F.dr
(2yi + x^2j).(dxi + dyj)= 2y.dx + x^2dy
Now what do I integrate it to? Let's assume they take 2 paths to reach (10,10) 1.(0,10) to (10,10) 2. (10,0) to (10,10). Now if i get different work done, they are non-conservative. But again, how do I integrate it? Is there a way to tell it's non-conservative without all this?