Let me first break this down in English from my understanding:

[itex]\nabla V[/itex] is the gradient of a scalar function [itex] V[/itex]. [itex]\nabla V[/itex] is a vector field at each point P where the vector points to the direction the maximum rate of increase and [itex]|\nabla V|[/itex] is the value of the slope.

[itex] \nabla \cdot \vec{A}[/itex] at a point P is the divergence of [itex] \vec{A}[/itex] at point P. If [itex] \nabla \cdot \vec{A}[/itex] at a point P is not zero, there must be a source or sink because the inflow to point P is not equal to the outflow from point P.

So what is the meaning of the divergence of a gradient ([itex]\nabla^2 V = \nabla \cdot \nabla V[/itex])?

What is the meaning of Laplace equation where [itex]\nabla^2 V = 0[/itex]?

What is the meaning of Poisson's equation where [itex]\nabla^2 V = [/itex] some function?

Please explain to me in English. I know all the formulas already, I just want to put the formulas into context.

Thanks

Alan