## Gauss’ Law, Part 1

In Derivation #3, the expression, $\frac{\partial \beta(t)}{\partial t}$, was written. This is an expression for a derivative of a function $\beta(t)$. Now that a derivative has been introduced, Maxwell’s equations can be investigated. I start with Gauss’ Law. But first, slightly more information about derivatives is needed. I can consider the pieces of $\frac{\partial \beta(t)}{\partial t}$ separately. These pieces are $\beta(t)$ and $\frac{\partial}{\partial t}$. The $\beta(t)$ is, of course, a function while $\frac{\partial}{\partial t}$ is an operator called a differential operator that forms $\frac{\partial \beta(t)}{\partial t}$ if $\frac{\partial}{\partial t}$ is applied to the left of the function $\beta(t)$. A differential operator can be made for each of the coordinates in a particular Cartesian coordinate system. For example, I can write $\frac{\partial}{\partial x}$ for the $x$-axis, $\frac{\partial}{\partial y}$ for the $y$-axis, and $\frac{\partial}{\partial z}$ for the $z$-axis. Next, define three unit vectors. Let $\vec{e}_x$ point along the positive $x$-axis, $\vec{e}_y$ point along the positive $y$-axis, and $\vec{e}_z$ point along the positive $z$-axis. Now there is a unit vector for each of three spatial dimensions. The differential operators and the unit vectors can be combined to form another operator called the del-operator, $\vec{\nabla}$. Definition 1 Define $\vec{\nabla} \equiv \frac{\partial}{\partial x} \vec{e}_x + \frac{\partial}{\partial y} \vec{e}_y + \frac{\partial}{\partial z} \vec{e}_z$. Equivalently, one can use the notation $\vec{\nabla} \equiv (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$. Now it is possible to write Gauss’ Law. Knowing what the end result should be is helpful for the upcoming derivation. It is $\vec{\nabla} \cdot \vec{E}(\vec{r}) = 4 \pi \rho(\vec{r}).$ In this equation, $\vec{r} \equiv (x,y,z)$ is the position vector, $\vec{E}(\vec{r}) \equiv (E_x(\vec{r}), E_y(\vec{r}), E_z(\vec{r}))$ is the electric field, and […]