Chapter 15.3, Problem 31ES

Prove: If $\text{F}\left(x,y,z\right)=f\left(x,y,z\right)\text{i}+g\left(x,y,z\right)\text{j}+h\left(x,y,z\right)\text{k}$ is a conservative field and f, g, and h are continuous and have continuous first partial derivatives in a region, then $\frac{\partial f}{\partial y}=\frac{\partial g}{\partial x},\frac{\partial f}{\partial z}=\frac{\partial h}{\partial x},\frac{\partial g}{\partial z}=\frac{\partial h}{\partial y}$ in the region.