Let $\displaystyle \bold K(x,y,z)=(4xyz,2x^2z+2yz^2,2x^2y+2y^2z+4z^3)$ defined in $\displaystyle \mathbb{R}^3$.

Show that the field is irrotational and calculate the general potential of this field.

I've calculated its curl (0), so the field is irrotational.

Now I must find, I believe, a vector field whose curl is $\displaystyle \bold K$. I called it $\displaystyle \bold A=(a_1,a_2,a_3)$ but I've too much unknowns. I don't know how to solve the last part. Can someone help me?

In details I wrote $\displaystyle \frac{\partial a_3}{\partial y}-\frac{\partial a_2}{\partial z}=4xyz$ and 2 other relations of the same kind. At last I ended up with at least 9 unknowns and only 3 equations.