Vector Fields in $\mathbb{R}^3$

Question

I just need help figuring out if it there is an example or if no such example exists.

Answer 1

Take $X=(x,xy,z)$. Suppose that there exists $f$ with $\partial_xf=x, \partial_yf=xy, \partial_zf=z$.

You must have:

$\partial^2_{xy}f=\partial_yx=0=\partial_xxy=y$. Contradiction.

Answer 2

Let $F\in C^1(\Bbb R^3)$. Then $F$ is the gradient of some $C^2$ function $f:\Bbb R^3\to \Bbb R$ if and only if $\operatorname{curl}(F)=0$ on all of $\Bbb R^3$.