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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.


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

Example 8.36  

Example 8.37  

Noah Dana-Picard