Using the solid angle to simplify an integral when deriving Gauss' Law

Homework Statement:

The following is a derivation of Gauss's Law using the solid angle.

Relevant Equations:

$$\Phi _e=\oint_{S}^{} \vec{E}\cdot d\vec{A}$$

I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.

haruspex
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Homework Statement:: The following is a derivation of Gauss's Law using the solid angle.
Relevant Equations:: $$\Phi _e=\oint_{S}^{} \vec{E}\cdot d\vec{A}$$

View attachment 270332View attachment 270333

I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.
It's fairly detailed. You will need to indicate where you think some steps have been missed.

cwill53
It's fairly detailed. You will need to indicate where you think some steps have been missed.
Taking a look at it again, I believe my issues can be chalked up to vector calculus weakness. I'm trying to work through it all, it will just take time. I will ask again if I'm still confused after I strengthen my calculus.