Proving Symmetry of Modified Maxwell Equations

[KFC]

If there were magnetic monopole, the Maxwell equations should be modified as

$$\nabla\cdot\vec{B} = \mu_0\rho_m$$

$$\nabla\times\vec{E} = - \frac{\partial \vec{B}}{\partial t} - \mu\vec{J}_m$$

and plus the other two.

I wonder how to prove these two modified equations are of correct form? Someone told me they should be of those form because of symmetry. I would like to prove that mathematically but have no way to start ...

 

[kof9595995]

Em,I think it's not possible by just derive theoretically and do not use the idea of symmetry,because Maxwell equations themselves are mathematical summaries of experimental laws,right?

 

[Meir Achuz]

The divB equation is the definition of "magnetic charge".
The curl E follows from the continuity equation for magnetic charge.

 

[Defennder]

"magnetic charge density", to be exact.