flux (in the plane)

How much a vector field crosses through a closed curve, outward or inward — the net amount of "stuff" the field carries across the boundary.

For $C$ parametrized by $\mathbf{r}(t)$ with outward unit normal $\mathbf{n}$ (obtained by rotating the unit tangent $\mathbf{T}$ by $-90°$ for a counterclockwise curve), flux is $\oint_C \mathbf{F}\cdot\mathbf{n}\,ds = \oint_C P\,dy - Q\,dx$ — the line integral of $\mathbf{F}$ dotted against the normal direction around a closed loop.

Examples

A whirlpool has huge circulation but can have zero net flux, because what spins in also spins back out.
For $\mathbf{F}=(x,y)$ around the unit circle: flux $=2\pi$ , a constant outward flow crossing the boundary.