Divergence
The divergence of a vector field \(\mathbf{F}\) is mathematically defined as the limit of the ratio of the surface integral of \(\mathbf{F}\) of the closed volume \(V\) enclosing \((x,y,z)\), where \(V\) shrinks to zero \[\begin{equation} \require{physics} \text{div}~\mathbf{F} % = \lim_{\Delta V \to 0} % \frac{1}{\Delta V} % \iint_S \mathbf{F} \vdot \mathbf{\hat{n}} \dd{S} % \thinspace . \end{equation}\] By arbitrary means, the divergence is positive if the flow is moving outwards. Since the closed volume \(V \to 0\), the divergence can physically be interpreted as the net flow of field out of an infinitesimal small volume and thus the extent to which the field diverges from a point. This quantity is clearly a scalar and thus the divergence is a vector operator that operates on a vector field, producing a scalar function of position. In Cartesian coordinates, the divercence of \(\mathbf{F}\) becomes \[\begin{equation} \text{div}~\mathbf{F} % = \boldsymbol{\nabla}\vdot\mathbf{F} % = \qty( \pdv{}{x},\pdv{}{y},\pdv{}{z} ) % \vdot (F_x,F_y,F_z) % = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z} % \thinspace . \end{equation}\]
Divergence theorem
For the sake of simplicity, a mathematically rigourous proof is not presented here, but can be found in diverse advanced calculus books. The divergence theorem states that \[\begin{equation} \iint_S \mathbf{F}\vdot\mathbf{\hat{n}}\dd{S} % = \iiint_V \boldsymbol{\nabla} \vdot \mathbf{F} \dd{V} % \thinspace . \end{equation}\] As may be seen, this theorem relates the surface integration with volume integration and states that the flux of a vector function through some closed surface equals to the divergence of that function over the volume enclosed by the surface (Schey 2005). But what does this mean physically? In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary, in this case the surface \(S\).