## Definiton of Divergence

FlimIn English, the Divergence of a vector field
$\mathbf{F}$
,
$\mathbf{\nabla} \cdot \mathbf{F}$
at a point is the flux per unit volume out of a volume
$\delta V \rightarrow 0$
containing the point, with surface
$S$

Mathematically,
$\mathbf{\nabla} \cdot \mathbf{F} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \mathbf{F} \cdot \mathbf{n} dS$

By the Divergence Theorem,
$\int \int \int_{\delta V} \mathbf{\nabla} \cdot \mathbf{F} dV = \int \int_S \mathbf{F} \cdot \mathbf{n} dS$

By the Mean Value Theorem for volumes,
$\int \int \int_{\delta V} \mathbf{\nabla} \cdot \mathbf{F} dV =(\mathbf{\nabla} \mathbf{F})_{(x_0 , y_0 , z_0)} \delta V \rightarrow (\mathbf{\nabla} \mathbf{F})_{(x_0 , y_0 , z_0)} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \mathbf{F} \cdot \mathbf{n} dS, \: (x_0 , y_0 , z_0) \in \delta V$