How can we describe the level curves?

Question

Using polar coordinates, I have to describe the level curves of the function:

$$ f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow \left\{\begin{matrix} \frac{2xy}{x^2+y^2} & \text{ if } (x, y) \neq (0,0)\\ 0 & \text{ if } (x, y)=(0, 0) \end{matrix}\right.$$

I have done the following:

$x=r \cos \theta , y=r \sin \theta$

If $(x, y) \neq (0, 0)$ then: $$\frac{2xy}{x^2+y^2} =\frac{2r \cos \theta r \sin \theta}{r^2 \cos^2 \theta+r^2 \sin^2 \theta} =\frac{2r^2 \cos \theta \sin \theta}{r^2}=2 \cos \theta \sin \theta=\sin 2 \theta$$

How can we describe the level curves??

Answer 1

The level curves of $f$ are the curves $f=constant$. In this case, $\sin2\theta=constant$. We can call the constant $\sin2\alpha$, where $-\frac12\pi\le2\alpha\le\frac12\pi$, and solving the equation $$\sin2\theta=\sin2\alpha$$ gives the level curves $$\theta=\alpha\ \hbox{or}\ \frac\pi2-\alpha$$ for $-\frac14\pi\le\alpha\le\frac14\pi$.