Let's say there's an $\textbf{f}$ for which $\mathbf{\triangledown} W = \textbf{f}$ . In addition $W(0,0,0)=0$ and the second partial derivative of $W$ is continuous. I am asked to find whether $W$ exists when $$\textbf{f}=(2x+yz)\mathbf{e_{x}}+(2y+zx)\mathbf{e_{y}}+(xy+1)\mathbf{e_{z}}$$
where $\mathbf{e_{x},\mathbf{e_{y}},\mathbf{e_{y}}}$ are orthogonal vectors which form a basis with the origin. If it does exist I have to give an example.
I'm not so sure what to exactly do in this case. I have found the second partial derivates of $\mathbf{f}$ but don't really know what to do with them. Any hints?
With $\frac\partial{\partial x}W=2x+yz$, you can find that $W=x^2+xyz+C(y,z)$, where $C$ is a function of $y$ and $z$. You can try to use the two other derivatives to find a solution and refine $C$. If there is one solution then there is an infinite amount of solutions (to an additive constant).