Theory

Introduction

We have already defined divergence in 3D:

$$\begin{array}{rl}\nabla \cdot \mathbf{F}& =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot (F_1,F_2,F_3)\\ & =\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.\end{array}$$

The scalar field $\nabla \cdot \mathbf{F}$ describes the rate of 3D expansion of $\mathbf{F}$ at each point in space. One can show this by integrating a vector integral across a small sphere of radius $r$, dividing by the volume of this sphere, and then taking the limit $r\to 0$.

The divergence theorem relates the 3D integral of $\nabla \cdot \mathbf{F}$ over the interior of a region of space $\mathcal{V}$ to the vector surface integral of $\mathbf{F}$ over the boundary $\mathcal{S}=\partial \mathcal{V}$ of the same region:

Divergence theorem

For a 3D region $\mathcal{V}$ with boundary $\partial \mathcal{V}$, we have:

$${\iiint }_{\mathcal{V}}\nabla \cdot \mathbf{F}dV={\iint }_{\partial \mathcal{V}}\mathbf{F}\cdot d\mathbf{A}.$$

As for its 2D version, this theorem says that when the elements of local expansion times volume, quantified as $\left(\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}\right)dV$, are added up across the full interior of $\mathcal{V}$, the result is equal to the total flux exiting $\mathcal{V}$ across its boundary $\partial \mathcal{V}$.

As before, to express the logic behind this theorem, one first divides the region $\mathcal{V}$ into small cubes, adds the expansion elements for each cube, and takes the limit as the cube size goes to zero. Flux across an internal face of a cube corresponds to flux exiting one cube and simultaneously flux entering another cube, so internal cancellation occurs. Only flux exiting the boundary $\partial \mathcal{V}$ (crossing an external face) is not cancelled and thus contributes to the final result.