What is the proof for the Vortex vector field equation?

Question

I'm Struggling to understand why the vortex vector field is given by:

Vortex vector field equation

$\vec F(x,y) = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$

If anyone could explain why this is, I would be very grateful.

Thank you.

Answer 1

For $\vec F=-\hat x \frac {y}{x^2+y^2} +\hat y \frac{x}{x^2+y^2}$, we see that

$$|\vec F|=\frac1r$$

where $r=\sqrt{x^2+y^2}$ is the polar coordinate for the magnitude of the position vector $\vec r=\hat r r$.

Moreover, the direction of $\vec v$ is the polar unit vector $\hat \theta$ and is perpendicular to the position vector.

Hence, $\vec F$ rotates (circulates) around the origin counterclockwise and its "strength" increases as we move closer to the origin.

A point of interest is that while $\nabla \times \vec F=0$ for all $\vec r\ne0$, the line integral of $\vec F$ is not $0$ for any (smooth) contour that encircles the origin.