Let's do a two dimensional case please: I guess what I am asking is this: Are there any gradients that are not generated by a potential function and what do these gradients generated by a potential function have to do with path independence ?
A gradient of a function generates many vectors, it is a place holder for values of $x$ and $y$ and when various values are inserted into $x$ and $y$ you generate a vector field. Now if there is a potential function that means there is an antiderivative for each of those place holders correct?
It also means the vector field is conservative which in turn means the path you take from one point in the field to another point in the field doesn't make any difference ... the energy you would end up gaining or loosing is the same.
But why does the existence of antiderivatives and potential functions mean that energy is conserved following a path through the field.? I used the example $\langle-y, x\rangle$ because what ever this link is then it would be missing for $\langle -y,x\rangle$ and perhaps I can see the difference such as $\langle x, y \rangle$ .
There is a fundamental link I am missing. I need some help.