Flux through a surface

Definition

Let $S \subseteq \mathbb{R}^3$ be an oriented surface with unit normal field $n$, and let $F$ be a continuous vector field on $S$. The flux of $F$ through $S$ is the surface integral $$\iint_S F \cdot n \, dS.$$

Remarks

If $r \colon D \to \mathbb{R}^3$ is a regular parametrisation of $S$ with $r_u \times r_v$ pointing in the direction of $n$, then the flux is $\iint_D F\big( r(u, v) \big) \cdot \big( r_u \times r_v \big) \, du \, dv$. Reversing the orientation of $S$ changes the sign of the flux. The notation $\iint_S F \cdot d\mathbf{S}$, with $d\mathbf{S} = n \, dS$, is common. Example: for $F(x, y, z) = (x, y, z)$ and the unit sphere with the outward normal $n(p) = p$, we get $F \cdot n = 1$, so the flux is the area $4\pi$. If $F$ is the velocity field of a fluid, the flux is the volume of fluid that crosses $S$ per unit time. Flow in the direction of $n$ counts as positive.

Explanations

L1 human

Flux measures how much of a field goes through a surface. Only the part of the field perpendicular to the surface counts. Flow along the surface does not cross it.