Maxwell's equation has well defined divergence?

[flux!]

Homework Statement

How to I explain that maxwell's equation has well defined divergence

Homework Equations

All four EM Maxwell's equation

The Attempt at a Solution

I discussed it by showing one of the property of Maxwell's equation that is the Divergence of a Gradient is always zero (With full solution of proof)

That is

∇⋅∇V = 0

where V is the potential of E-field and

∇⋅∇A = 0

where A is the potential of the B-field

However, it seams that this solution does not satisfy our Professor, now I don't know how to answer this question since by definition when a function is said to be well defined then it should obey its property.

 

[Charles Link]

I'm not sure what the professor is looking for, but $-\nabla \cdot \nabla V=\rho/\epsilon_0$ where $\rho$ is the electric charge density. Two vector identities that are always the case are divergence of curl equals zero and curl of gradient equals zero. The divergence of the gradient is not always equal to zero. Maxwell's equations have $\nabla \cdot B=0$ and $\nabla \cdot E=\rho/\epsilon_o$, but I don't know exactly what your professor is looking for. Perhaps he is looking for the result that it (the E field and the B field) obeys Gauss's law because it has a well defined divergence but that is just a guess...