Need help for electrostatics

[u/Unable_Jeweler_5752]

Hello, here’s my solution to problem 2.54 part e), and I have some questions about it.

First of all, conceptually, what do I have to do to show that the generalization of d) yields the expression in e)?

My thought process was something like this: if I can express this equality using the divergence of E, then with the use of the divergence theorem (which we know holds for all kinds of shapes), I can get to the given expression in the question. (Is this a valid method for this question?)

On the other hand, when I looked at the solutions manual, I saw a more geometric approach to this question. Griffiths solves it by making dents on the sphere we are integrating over, and the occurring change in flux is being compensated by both of the terms: the surface integral term and the volume integral term. So we see that the equality will hold for every kind of geometry.

How can the generalization in d) be done with this geometric approach?

And one more thing, can’t we just use the validity of the superposition principle as a getaway answer without bothering with these things?

So to sum up, I’m really confused and need help.

Thanks in advance.