flux integral

The 3D surface analogue of flux in the plane, $\iint_S\mathbf{F}\cdot d\mathbf{S}$ , measuring how much a vector field flows through an oriented surface.

A flux integral is $\iint_S\mathbf{F}\cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v))\cdot(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv$ , dotting the field against the unnormalized normal direction — a field component parallel to the normal contributes fully, a field lying entirely in the tangent plane contributes nothing. Reversing the surface's orientation flips the sign of $\mathbf{r}_u\times\mathbf{r}_v$ and therefore of the whole integral.

Examples

For $\mathbf{F}=\langle 0,0,1\rangle$ through the cone $\mathbf{r}(u,v)=\langle u\cos v,\,u\sin v,\,u\rangle$ , oriented outward: $\iint_S\mathbf{F}\cdot d\mathbf{S} = 4\pi$ .
This integral is the left-hand side of Gauss's Law in electromagnetism.