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Flux across an oriented surface.

Suppose that $\overrightarrow{F} $ is a vector field, continuous in a region of the space which contains the oriented surface $\mathcal{S}$ (v.s. subsection  4.5).

Definition 8.33   The flux of $\overrightarrow{F} $ across $\mathcal{S}$ is the integral of $ \overrightarrow{F}\cdot
\overrightarrow{n} $ over $\mathcal{S}$, where $\overrightarrow{n} $ is the chosen unit normal vector.

\begin{displaymath}\text{Flux} = \iint_{\mathcal{S}} \overrightarrow{F}\cdot \overrightarrow{n}\; d \sigma.

Example 8.34  

Noah Dana-Picard