How to deduce Gauss' law from Gauss Divergence Law

[nenyan]

Homework Statement

Gauss Divergence Law:

Gauss' law

Can we obtain the Gauss' Law from Gauss Divergence Law?

Homework Equations

In Spherical coordinates,

electric field strength

(Q/4$\pi$εr^2,0,0)
Then ∇$\cdot$E=0+0+0=0

The Attempt at a Solution

We can not obtain the Gauss' Law from the general mathematical law?

 

[UltrafastPED]

So the volume integral of div E [application of Gauss' theorem to Gauss' law) is proportional to Q, the total charge included.

But Q is the volume integral of the charge density ... so we have a volume integral on both sides.

Then the integrands must be equal, which gives div E = densit/eps_0.

 

[vela]

Homework Statement

Gauss Divergence Law:

Gauss' law

Can we obtain the Gauss' Law from Gauss Divergence Law?

Homework Equations

In Spherical coordinates,

electric field strength

(Q/4$\pi$εr^2,0,0)
Then ∇$\cdot$E=0+0+0=0

The Attempt at a Solution

We can not obtain the Gauss' Law from the general mathematical law?

There's a singularity in $\vec{E}$ at r=0 you need to account for.