# Math Insight

### Polar coordinates mapping

Polar coordinates can be viewed as a way to determine the location of a point on the plane via the coordinates $(r,\theta)$, as described on another page and illustrated with the following applet.

Polar coordinates. When you change the values of the polar coordinates $r$ and $\theta$ by dragging the red points on the sliders, the blue point moves to the corresponding position $(x,y)$ in Cartesian coordinates. Alternatively, you can move the blue point in the Cartesian plane directly with the mouse and observe how the polar coordinates on the sliders change. The coordinate $r$ is the length of the line segment from the point $(x,y)$ to the origin and the coordinate $\theta$ is the angle between the line segment and the positive $x$-axis.

In the above applet, the polar coordinates $r$ and $\theta$ were viewed as two separate numbers, which you could change using two points on separate one-dimensional lines (the sliders). We can change our perspective slightly by representing the polar coordinates as single point on a two-dimensional polar $(r,\theta)$ plane. After all, polar coordinates $(r,\theta)$ are just an ordered pair of numbers, like Cartesian coordinates, so we can define an $r$-axis and a $\theta$-axis in direct analogy to the typical $x$-axis and $y$-axis.

With this perspective, polar coordinates is a mapping from a point $(r,\theta)$ in the polar coordinate plane to the correspoing point $(x,y)$ in the Carteisan coordinate plane. The following applet is virtually identical to the above applet, but it illustrates this new perspective by allowing you to specify the polar coordinates using a single point in the $(r,\theta)$ plane.

Polar coordinates with polar axes. The red point in the inset polar $(r,\theta)$ axes represent the polar coordinates of the blue point on the main Cartesian $(x,y)$ axes. When you drag the red point, you change the polar coordinates $(r,\theta)$, and the blue point moves to the corresponding position $(x,y)$ in Cartesian coordinates. Alternatively, you can move the blue point in the Cartesian plane directly with the mouse and observe how the polar coordinates represented by the red point change. The coordinate $r$ is the length of the line segment from the point $(x,y)$ to the origin and the coordinate $\theta$ is the angle between the line segment and the positive $x$-axis.

This new perspective allows us to explore the features of this mapping. Using the conversion formulas for polar coordinates, \begin{align*} x &= r\cos\theta\notag\\ y &= r\sin\theta, \end{align*} we can write the mapping as the transformation $\vc{T}$ defined by $(x,y)=\cvarf(r,\theta) = (r\cos\theta,r\sin\theta)$. In the above applet, we mapped the point $(r,\theta)$ into the point $(x,y)=\vc{T}(r,\theta)$. We can gain additional intuition into the behavior of the polar coordinates mapping $\vc{T}$ by looking at how it transforms sets of points.

For example, we can explore how polar coordinates maps a rectangle in the $(r,\theta)$ plane. If a rectangle $\dlr^*$ is determined by $a \le r \le b$ and $c \le \theta \le d$, it is mapped into what kind of object $\dlr$ in the Cartesian plane? Given that $$x^2 +y^2 = r^2,$$ the constraint $a \le r \le b$ means that $a^2 \le x^2+y^2 \le b^2$, i.e., that the points $(x,y)$ must lie in an annulus with inner radius $a$ and outer radius $b$. The constraint $c \le \theta \le d$ means that the points $(x,y)$ must lie in a sector of the $xy$ plane spanning the angle $c \le \theta \le d$ with the positive $x$-axis. You can explore the nature of the resulting region $\dlr$ using the following applet. If you use the full range $0 \le \theta \le 2\pi$ and a range $0 \le r \le b$, then you will see that the rectangle $\dlr^*$ is mapped into the disk $\dlr$ of radius $b$.

Polar coordinates map of rectangle. The transformation from polar coordinates to Cartesian coordinates $(x,y)=\cvarf(r,\theta) = (r \cos\theta, r \sin \theta)$ can be viewed as a map from the polar coordinate $(r,\theta)$ plane (left panel) to the Cartesian coordinate $(x,y)$ plane (right panel). This transformation maps a rectangle $\dlr^*$ in the $(r,\theta)$ plane into a region $\dlr$ in the $(x,y)$ plane that is the part of an angular sector inside an annulus. You can change the regions $\dlr^*$ and $\dlr$ by dragging the purple or cyan points in either panel. To further visualize the action of the map $(x,y)=\cvarf(r,\theta)$, you can drag the labeled red and blue points anywhere inside the large rectangle $0 \le r \le 6$, $0 \le \theta <2\pi$ and corresponding disk $x^2+y^2 \le 6^2$.

This perspective of polar coordinates as a mapping allows us to do even more: we can look at how $\vc{T}(r,\theta)$ changes the area of regions as it maps from the $(r,\theta)$ plane to the $(x,y)$ plane. The following applets illustrates how the stretching or shrinking of $\vc{T}$ depends on the location $(r,\theta)$. You can observe how it stretches more as $r$ increases, shrinking area substantially for very small $r$.

Area transformation of polar coordinates map. The transformation from polar coordinates to Cartesian coordinates $(x,y)=\cvarf(r,\theta) = (r \cos\theta, r \sin \theta)$ maps a rectangle $\dlr^*$ in the $(r,\theta)$ plane (left panel) to the region $\dlr$ in the $(x,y)$ plane (right panel). It also maps each small rectangle in $\dlr^*$ to a “curvy rectangle” in $\dlr$. Although the small rectangles in $\dlr^*$ are the same size, the corresponding “curvy rectangles” vary greatly in size. Depending on the coordinates $(r,\theta)$, the map $\cvarf(r,\theta)$ shrinks or expands the area by different amounts. You can visualize the mapping of the small rectangles by dragging the yellow point in either panel; the corresponding small rectangle in $\dlr^*$ and its image in $\dlr$ are highlighted. You can also change the regions $\dlr^*$ and $\dlr$ by dragging the purple and cyan points in either panel.

#### Keeping track of changes in area

It turns out that the stretching or shrinking of area by mappings like polar coordinates plays a big role in multivariable calculus. For linear transformations, this stretching or shrinking is particularly easy to visualize as it always stretches or shrinks by the same amount. For nonlinear transformations like polar coordinates, we saw that the stretching or shrinking can change depending on location. A method for capturing how the stretching or shrinking changes is a central component of changing variables in double integrals.