Thank you, the condition $\displaystyle \frac{di}{dy} - \frac{dj}{dx} = 0$ is correct ?
it shouldn't be i and j there, but you have the right idea, if $\displaystyle \bold{F} = P(x,y)i + Q(x,y)j$, then we can find a potential function if $\displaystyle \frac {\partial P}{\partial y} = \frac {\partial Q}{\partial x}$
To confirm learning
$\displaystyle F(x,y) = (x^2y)i + (5xy^2)j$
Field not is gradient (conservative) because $\displaystyle \frac{d(x^2y)}{dy} \not= \frac{d(5xy^2)}{dx}$
it should be partial derivatives: $\displaystyle \frac{\partial (x^2y)}{\partial y} \not= \frac{\partial (5xy^2)}{\partial x}$, so it is not conservative
I have one problem with the same F but the question is: Field is gradient ?
It is the same as: field is conservative?
i think you are describing the same thing, but it probably would not be stated like that. saying the field is "conservative" is saying it is the "gradient function of some vector field".