If $F$ is a velocity field, then the integral $\iint_S\vec{f}\cdot\vec{n}dA$ represents the total flow of the field $F$ through the surface $S$. Why is that integral and integrand the correct choices to calculate that quantity?
2026-03-25 16:54:57.1774457697
flow of velocity field
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Intuitively, you would expect the flux to be highest when the vector field is perpendicular to the surface and to be zero if it is parallel to the surface, as nothing actually flows through the surface in the latter case. This explains the dot product of the vector field $f$ and the surface normal unit vector $\hat n$.
To get the total flux through the surface, we look at every point on the surface, calculate the flux through that point using the dot product for the reason specified above, and then sum it all up to get the total flux, which is precisely what the integral does, informally.