Given a continuous vector field $\vec{F}$ over a region $R$, let $C$ be a smooth oriented curve in $R$. The outward $\textbf{flux}$ of $\vec{F}$ across $C$ is:

$$\int_C \vec{F}\cdot\vec{n} ds$$

...where $\vec{n}$ is the unit outward normal vector.

Let $C$ be the counterclockwise oriented unit circle on the plane and centered at the origin.

Find the flux of $\vec{F}=\langle x,y\rangle $ on $C$.