Are we really ever plotting in polar coordinates?

Question

Is it true that when we plot in polar coordinates, we are still actually plotting in the $x$-$y$ coordinate system? Wouldn't plotting in the polar coordinate system really be plotting with $\theta$ and $r$ on the orthogonal axes?

Thank you

Answer 1

Yeah, you always plot in the $xy$ plane.

When you say you are using polar coordinates, it means that the free variables are $\theta$ and $r$, so you express $x$ and $y$ in function of those two.

Answer 2

It is perfectly possible to plot in "real" polar coordinates, with $r$ and $\theta$ on what seem to be orthogonal axes. I do this all the time when I teach calculus. For example, if we draw the region in polar coordinates determined by $0 \leq r \leq 1$ and $0 \leq \theta \leq 2\pi$, we obtain what looks like a rectangle. This is why it is so easy to set up an the integral to find the area of the unit circle using polar coordinates.

This sort of diagram is alsu helpful to explain the role of the Jacobian in setting up such integrals; the Jacobian shows the local "stetch rate" of area between the region when it is mapped from polar coordinates to Cartesian coordinates by the map $(r,\theta)\mapsto (r \cos \theta, r \sin \theta)$. This is where the $r$ in $dx\,dy = r\,dr\,d\theta$ comes from.

It is also useful, in the context of multivariable calculus, to plot the shadow of various volumes in "orthogonal" $z/r$ coordinates (ignoring the $\theta$ coordinate of cylindrical coordinates). In this sort of diagram, the projection of a solid cylinder also looks like a rectangle.

Answer 3

The plane is the plane. There are many ways to coordinatize (specify) points.