The coordinate system

You should be familiar with polar coordinates from trigonometry. Here, the coordinates of a point in the plane are expressed by r and $\theta$ where r is the distance of the point from the origin, and $\theta$ is the angle (in radians) between the x-axis and a line segment joining the origin and the point.

A simple bit of algebra will show that

$$x = r\cos \theta, \quad y = r\sin \theta, \quad \mbox{and, thus,}$$

$$r = \sqrt{x^2 + y^2}, \quad \theta = \arctan \left(\frac{y}{x}\right).$$

So, if we have a function in Cartesian coordinates that is f(x,y) = x² + y², we can write this in polar coordinates as $f(r,\theta) = r^2$.

Notice that the curves r = r₀ describe circles of radius r₀ in the xy-plane, while the lines $\theta = \theta_0$ describe straight lines through the origin at an angle $\theta_0$ from the x-axis.