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,
\[\int \int \int_{\delta V} \mathbf{\nabla} \cdot \mathbf{F} dV =(\mathbf{\nabla} \mathbf{F})_{(x_0 , y_0 , z_0)} \delta V \rightarrow (\mathbf{\nabla} \mathbf{F})_{(x_0 , y_0 , z_0)} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \mathbf{F} \cdot \mathbf{n} dS, \: (x_0 , y_0 , z_0) \in \delta V \]