Total Electric Flux through Surface

Physical Law
Let $\mathbf E$ be an electric field which acts on a region of space $R$.

Let $S$ be a surface embedded in $R$.

The total electric flux through $S$ is the surface integral over $S$ of the dot product of $\mathbf E$ with the vector area of $S$:


 * $F = \ds \int_S \mathbf E \cdot \rd \mathbf S$

where $\d \mathbf S$ is an infinitesimal area element of $S$.

Proof
Let $S$ be divided up into area elements of vector areas $\delta \mathbf S$, where the sign of each area element is defined in the same sense.

The electric flux through $\delta \mathbf S$ is defined as:
 * $\mathbf E \cdot \delta \mathbf S$

Summing all of these, we get:
 * $F = \ds \sum_{\text {all surfaces $\delta \mathbf S$} } \mathbf E \cdot \delta \mathbf S$

As $\ds \lim_{\text {area of $\delta \mathbf S$} } \to 0$, this becomes the surface integral over $S$:
 * $F = \ds \int_S \mathbf E \cdot \rd \mathbf S$