The Divergence theorem.

The divergence of a plane vector field has been already defined, in section 6.

Theorem 8.35   Let $\overrightarrow{F} =M \overrightarrow{i} + N \overrightarrow{j} + P \overrightarrow{k}$be a vector field defined over a domain in space.

Let $\mathcal{S}$be a closed oriented surface which encloses the region $\mathcal{D}$in space; we denote $\overrightarrow{n}$a unit vector, normal to the surface $\mathcal{S}$and pointing outwards.

Then:

$$\iint_{\mathcal{S}} \overrightarrow{F}\cdot \overrightarrow{n... ...hcal{D}} \overrightarrow{\nabla}\cdot \overrightarrow{F}\; dV.$$

Example 8.36  

Example 8.37