## Overview of Gauss’ Theorem and Basics of Integration

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 […]