Flux measures how much of a vector field passes through an oriented surface.

If $S$ has a chosen unit normal $\mathbf n$, the flux of $\mathbf F$ through $S$ is

$$\Phi=\iint_S\mathbf F\cdot\mathbf n,dS.$$

The dot product selects the component of the field perpendicular to the surface.

Tangential field does not cross the surface

If $\mathbf F$ is tangent to the surface, then

$$\mathbf F\cdot\mathbf n=0,$$

so it contributes no flux. If the field points in the normal direction, its full magnitude contributes.

For a constant field perpendicular to a flat surface of area $A$,

$$\Phi=\mathbf F\cdot\mathbf n,A.$$

Orientation controls the sign

A surface has two possible normal orientations. Reversing the normal gives

$$(-\mathbf n)$$

and therefore reverses the flux:

$$\Phi_{-\mathbf n}=-\Phi_{\mathbf n}.$$

For a closed surface, the standard orientation is outward.

Computing from a parametrization

If

$$\mathbf r(u,v)$$

parametrizes the surface with the chosen orientation, then

$$\mathbf n,dS =(\mathbf r_u\times\mathbf r_v),du,dv,$$

so

$$\iint_S\mathbf F\cdot\mathbf n,dS

\iint_D \mathbf F(\mathbf r(u,v))\cdot (\mathbf r_u\times\mathbf r_v),du,dv.$$

Flux is the oriented counterpart of a scalar surface integral and is central to field laws in fluid mechanics and electromagnetism.