I saw from "Advanced Engineering Mathematics, 10th Edition" by Kreyszig, p. 400, that the solution $V$ of the Laplace's equation,
$$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2} = 0,$$
is a potential function.
However I thought I can manipulate the expression as $\nabla^2V=0 \ \rightarrow \ \nabla \cdot (\nabla V)=0$. Because $\vec{F}=-\nabla V$, I can write the expression as $\nabla \cdot \vec{F} = 0$.
I did an example calculation with the gravitational force:
$$\vec{F} = - \frac{GMm}{r^2} \hat{r} = - \frac{GMm}{(x^2+y^2+z^2)^{3/2}} \ (x, y, z)$$
and I got $\text{div} \ \vec{F} = \vec{0}$.
But why is this not introduced in most mathematics / physics books?
Or is there any exception?
