It turns out that deriving Gauss’ Law is easier said than done. There are several steps according to a StackExchange post [1]. The first of these steps is understanding Gauss’ Theorem. Hmm. Perhaps Gauss used his own theorem to derive his electrostatics law.
After a quick online search, it is clear that Gauss’ Theorem is just another name for the Divergence theorem [2]. The Divergence Theorem is [3]
$ \int\int\int_V (\vec{\nabla} \cdot \vec{F}) dV = \int\int_{S} (\vec{F} \cdot \vec{e}_{n}) dS $.
Oof. That is a lot of symbolism to break down. Fortunately I am able to break this down; it will just take a while.
First, basics. The $ \int$ symbol is a fancy way to say “find the area formed between a function’s curve and one axis for a variable, as the function varies with respect to that variable.” Any variable with a $ d$ in front of it is called a dummy variable. For a given $ \int$ symbol, the dummy variable indicates the variable with which the function changes while the integral is being determined. It is easy to get caught up in attempting to interpret a dummy variable as the width of a shrinking shape. The dummy variable does provide the correct dimensions, however.
If there is more than one $ \int$ symbol, the function can vary with respect to more than one variable. In the above case, the expression on the left side includes three $ \int$ symbols as $ \int \int \int$, which indicates that there are three variables or dimensions to consider when evaluating. Since there are three $ \int$ symbols, there are three dummy variables. There is one dummy variable for each integral symbol: $ dV = dx dy dz$ in Cartesian coordinates. Multiplying three perpendicular lengths along the Cartesian axes produces a volume, $ V$. The triple integral has a subscript of $ V$ to indicate that a volume called $ V$ is involved in the integration process.
A similar situation occurs when there are two $ \int$ symbols, as is the case on the right side of the Divergence theorem. Multiplying two perpendicular lengths along Cartesian axes produces an area or a surface. The double integral has a subscript of $ S$ to indicate the surface involved in the integration. Note also that the dummy variable $ dS$ equals $ dx dy$ in the $ x-y$ plane. For the Divergence Theorem to apply, $ S$ must be a closed surface meaning that $ S$ is finite and has no boundary [4].
At a certain point in the domain of $ \vec{F}$, $ \vec{F}$ becomes a vector in the same way that at a certain point in the domain of a variable that variable becomes a number. The variable is not called a number just because it can be a number. Analogously, the vector field is not called a vector.
The collection of symbols to the right of $ \int$ symbols is called an integrand. The integrand on the left side consists of the dummy variable $ dV$ and the function $ (\vec{\nabla} \cdot \vec{F})$. Here, $ (\vec{\nabla} \cdot \vec{F})$ is the divergence of a vector field $ \vec{F}$.
The integrand on the right side of the Divergence Theorem is $ \vec{F} \cdot \vec{e}_n$. Here, $ \vec{e}_n$ is a unit vector that is perpendicular to the surface at every point on the volume $ V$. This vector points outward instead of inward towards the inside of the volume $ V$ [3].
To be continued…
References
[1] https://physics.stackexchange.com/questions/38404/how-is-gauss-law-integral-form-arrived-at-from-coulombs-law-and-how-is-the [2] https://mathworld.wolfram.com/DivergenceTheorem.html [3] https://en.wikipedia.org/wiki/Divergence_theorem [4] https://en.wikipedia.org/wiki/Surface_(topology) [5] https://math.stackexchange.com/questions/368155/laplacians-and-dirac-delta-functions
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