How do I find a function $f(x,y)$ such that $\nabla f = \langle y,-x\rangle$?

Question

I've been stuck on this problem for a while and I am starting to think that it's not possible. Could someone please point me in the right direction? Thanks!

Answer 1

If there were such a function $f$, you'd have $$\frac{\partial f}{\partial x} = y \quad\text{and}\quad \frac{\partial f}{\partial y} = -x.$$ Do you know something about $\dfrac{\partial^2 f}{\partial x\partial y}$ and $\dfrac{\partial^2 f}{\partial y\partial x}$?

Answer 2

The second partial dervitives $f_{xy}, f_{yx}$ are not equal so there is no twice differentiable function $f$ with this property.