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Let $\mathbf V$ be a vector field which acts on a region of space $R$.

Let $S$ be a surface embedded in $R$.

The total flux through $S$ is the surface integral over $S$ of the dot product of $\mathbf V$ with the vector area of $S$:

$F = \ds \int_S \mathbf V \cdot \rd S$

where $\d S$ is an infinitesimal area element of $S$.

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