Divergence theorem is green's theorem extended to oriented surfaces and vector fields in R3. The rate of change of amount of fluid inside a solid Q bounded by a surface $\partial Q$ can be computed in two ways. One way, it can be viewed as the flow into or out of Q across the boundary $\partial Q$ which is given by the Flux integral $\int \int _{\partial Q}F . ndS$. On the other hand, it can be considered as the sum of collection or dispersal of fluid at each point inside Q, which is given by the triple integral $\int \int \int _{Q}\bigtriangledown . F(x,y,z)dV$

Divergence Theorem

We thus, get an extension of Green's theorem known as Gauss's Divergence theorem.
$\int \int _{\partial Q}F.ndS$ = $\int \int \int _{Q}\bigtriangledown .F(x,y,z)dV$

Suppose Q is a simple solid region bounded by the surface $\partial Q$ and n(x,y,z) denotes the outward normal vector of $\partial Q$. Let F(x,y,z) be a vector field, whose components have continuous partial derivatives in an open region containing Q. Then,
$\int \int _{\partial Q}F.ndS$ = $\int \int \int _{Q}\bigtriangledown .F(x,y,z)dV$

To put the theorem in other words, it can be stated that the Flux integral of a continuously differentiable vector field across a boundary surface is equal to the integral of divergence of that vector field within the region enclosed by the boundary.

Using the divergence theorem, it can be proved, that the divergence of a vector field F at an interior point P0 is equal to the limiting value of the flux per unit volume over a sphere centered at P0, as the radius of the sphere tends to zero.

div F(P0) = $\lim_{a\rightarrow 0}\frac{1}{V_{a}}\int \int _{S_{a}}F.ndS$