Definition:Flux
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Definition
Let $\mathbf V$ be a vector field which acts on a region of space $R$.
Let $S$ be a surface embedded in $R$.
The flux through $S$ is the dot product of $\mathbf V$ with the vector area of $S$.
Total Flux
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 $S$.