So Im struggling with this question for a long time,hence thought posting it here might help.
The question that I have is about the Divergence of a vector field.
According to wikipedia - Divergence at particular POINT tells about how the imagined fluid flows in or out of small regions near it. This means we measure divergence at a point and not in a volume.
Now according to Maxwell's 1st equation $$\nabla \cdot E =q/\epsilon_0,$$ this means Divergence of a field at a point is equal to the charge at that point divided by some constant. So according to this equation, divergence at source point/positive charge point is non zero, at negative charge/ sink point is not zero, BUT is ZERO at ALL other points.
Contradicting this, when we calculate divergence by vector calculus, it comes non zero at other point, reason being there is a change in fields value happening around that point. Here is an example from the video - 3blue1brown explaining divergence
So, which one is correct? How can both be mathematically correct at the same time?
We have the equation $\nabla \cdot E = \frac{q}{\epsilon_0}$. Here $q$ is the electric charge density at some point rather than the electric charge. It might help to think more "macroscopically" where charge is not just concentrated in a point but spread throughout a volume. There is a formal way to consider point charges but it is more technical (because if all the charge is at one point, then the charge density is infinite in some sense).
The way to think it terms of charge rather than charge density is to use the integral form of Gauss's law: $$\int_\Omega \frac{q}{\epsilon_0} dV = \int_\Omega \nabla \cdot E \ dV =\int_{\partial \Omega} E \cdot dS$$ On the left is the total charge in the volume. If you take $\Omega$ to be a tiny volume around your point charge, then you can think of this whole integral as describing the effect of your electron or whatnot. By Maxwell's equations, it's equal to the middle expression, the integral of the divergence over the volume. The RHS is the total amount of electric flux through the boundary, which is equal by the divergence theorem. You might find this helpful for some diagrams.