# Divergence Theorem

## Table of Contents

If \(\small \vec{F}\) is a continuously diffrerentiable vector field in the volume \(\small V\) with closed surface \(\small S\), then the divergence theorem (also known as Gauss’s Theorem) states that

$$ \iiint_V (\nabla \cdot \vec{F}) dV = \oiint_S (\vec{F} \cdot \hat{n}) dS $$

Where

- The left hand side is a volume integral of the divergence of \(\vec{F}\) over the volume \(V\), and
- \(\hat{n}\) is the unit normal vector to the surface S, used in the surface integral on the right hand side.

This can be interpreted intuitively. Since the divergence of a vector field can be thought of as the degree to which a field is acting as a source or sink, the divergence theorem can be thought of as saying that the total amount of a field that is acting as a source or sink in a volume is equal to the amount of the field leaving or entering the volume through the surface.