Showing that a vector field $\vec{G} = (F_2(y, x), F_1 (y, x)$ is conservative given that $\vec{F} = (F_1 (x, y), F_2 (x, y))$ is conservative

Question

The question is in the title. I have tried a huge amount of counter-examples and have come to the conclusion that the vector field is conservative.

It can be shown by counter-example that the vector fields $\vec{G} = (F_1 (y,x), F_2 (y, x)$, and $(F_2 (x,y), F_1 (x, y)$ are not conservative. In essence the vector field in the question is a combination of these.

I have tried to directly compute the derivatives of the components of G but I always end up writing tautologies which do not prove anything.

I have also tried to generate some matrices which I could multiply G by to transform it, but I doubt there is a matrix that exists which would flip the internal components of the F components.

Any hints or suggestions as to how to continue?

Answer 1

Hint: If $F$ is conservative, then there is a $\phi$ so that

$$ F_1(x,y) = \frac{\partial}{\partial x} \phi(x,y)\\ F_2(x,y) = \frac{\partial}{\partial y} \phi(x,y)\\ $$

Let $\psi(x,y) = \phi(y,x)$. What are $\dfrac{\partial}{\partial x}\psi(x,y)$ and $\dfrac{\partial}{\partial y}\psi(x,y)$?