## Calculus (3rd Edition)

$$\phi (x,y,z) = xy z^2+K$$
Given $$\mathbf{F}=\left\langle y z^{2}, x z^{2}, 2 x y z\right\rangle$$ We need to find $\phi (x,y )$ such that $$\frac{\partial \phi }{\partial x}=y z^2,\ \ \ \frac{\partial \phi }{\partial y}=x z^{2},\ \ \ \ \frac{\partial \phi }{\partial z}=2x y z$$ By integration we get \begin{align*} \phi (x,y,z)&=xy z^2+C_1\\ \phi (x,y,z)&= xy z^2 +C_2 \\ \phi (x,y,z)&=xy z^2 +C_3 \end{align*} Then we can choose $$\phi (x,y,z) = xy z^2+K$$