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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

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  • $\begingroup$ You can plot a ray to infinity (the $r$ axis) and draw some circles around its starting point. Then you're looking at a polar coordinate chart of some sorts. $\endgroup$
    – Pedro
    Sep 21, 2014 at 2:21

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

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    $\begingroup$ This is a plausible, and perhaps even common, viewpoint. However, the points $(1,5)$ and $(1,5+2\pi)$ are different points on the polar plane in $(r,\theta)$ coordinates, but they map to the same point in the $(x,y)$ plane under the map $(x,y) = (r\cos\theta,r\sin\theta)$. So the $(r,\theta)$ plane is not quite the same as the $(x,y)$ plane from another viewpoint. $\endgroup$ Sep 21, 2014 at 2:12
  • $\begingroup$ Isn't that just a plane that you've coordinatized with a rectangular system and called the coordinates $r$ and $\theta$? $\endgroup$
    – paw88789
    Sep 21, 2014 at 3:07
  • $\begingroup$ Isn't that what "polar coordinates" means - a map from one plane with "rectangular" coordinates to another? $\endgroup$ Sep 21, 2014 at 3:10
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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.

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  • $\begingroup$ Ok, yeah I do remember seeing its use for things like the Jacobian and surface area. However, at least at the lower level, like precalc, people tend to say we are plotting in polar coordinates when they are not plotting in the "real" polar coordinates. And that can cause confusion with things like the gradient. $\endgroup$
    – dylan7
    Sep 21, 2014 at 1:57
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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.

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  • $\begingroup$ That's what I always thought. I feel like teachers leave that point. Thank you. $\endgroup$
    – dylan7
    Sep 21, 2014 at 1:07

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