Polar, Cylindrical, and Spherical Coordinates

We have included a comprehensive treatment of polar coordinates here because many students will not have a lot of experience with polar coordinates and because these ideas of angular coordinate measurements are fundamental to cylindrical and spherical coordinates in three dimensions. The preview activity is designed to help student use right triangle side and angle relationships outside of the rigid settings used in trigonometry. We do not anticipate that instructors will use every example or activity in this section, but we have offered a broad selection of ideas covered to allow instructors to tailor this material to the experience and needs of their students.

A traditional approach done in other texts involves waiting to introduce cylindrical and spherical coordinates until multiple integration. We have opted to include this material here to further develop students thinking about paths, surfaces, and regions having angular symmetry about an axis (cylindrical) or a point (spherical). This thinking is not strictly required but the authors have found this useful when parameterizing paths in 3D (in ), translating geometric features to algebraic forms (throughout the rest of this text), and with evaluating limits of multivariable functions (), as well as many other places where rotational/angular symmetry or descriptions occur.

For each of the coordinate systems in this section, there is discussion about how to convert points, equations, and regions between coordinate systems with a description of the geometric meaning of the coordinate measurements and a description for the algebraic conversion. Some of these conversions may seem simplistic while others can involve more tedious algebra. We encourage you to select activities that best connect your students' prior experience with which coordinate descriptions you want to emphasize later.

In this preview activity, we will investigate some measurements that will be useful when defining new ways to specify locations. Many of these measurements involve measuring the length of specific line segments and angles between line segments in three dimensions.

In this section we will define and work with new ways to specify the location of a point. These new measurements will be useful as coordinate systems and as a way of thinking about how to describe different shapes relative to rotational or other symmetries.

We previously defined the rectangular coordinates of a point to be signed distances from the point to the axes as shown below. Remember that the sign on the coordinate tells you whether to go above or below the other axis.

We define the polar coordinates of a point \(P\) in two dimensions to be \((r,\theta)\), where \(r\) is the signed distance from the origin to \(P\) and \(\theta\) is the counterclockwise angle from the positive horizontal axis to the line segment connecting the origin and \(P\). We use the sign on the \(r\) to indicate if the point is locked ahead or behind when facing the \(\theta\) direction at the origin.

A helpful way to visualize a location based on polar coordinates is to stand at the origin, facing the positive horizontal axis; turn an angle of \(\theta\) counterclockwise; and move \(r\) in the direction you are facing.

If you look at , you can see a right triangle in the first quadrant with base \(x\) and height \(y\). The angle the hypotenuse makes with the base is \(\theta\). From this, you can deduce the four relationships shown below, which are used to convert between rectangular and polar coordinates. It may be insightful to draw triangles of this type in each of the quadrants to confirm for yourself that these conversion formulas are valid in all four cases.

The following activity gives you an opportunity to practice converting between rectangular and polar coordinates.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

So far in this chapter, we have encountered two different coordinate systems in \(\R^2\): rectangular and polar coordinates. Earlier in this section, we saw how polar coordinates form a convenient alternative in certain situations. In a similar way, there are two more coordinate systems in \(\R^3\) that come from different ideas of rotational measurement. This subsection introduces cylindrical coordinates, which we can view as a vertical extension of polar coordinates to three dimensions. The next subsection will introduce spherical coordinates, which are useful for situations with significant rotational symmetry with respect to the origin.

Cylindrical coordinates are a coordinate system for \(\mathbb{R}^3\) that consists of using polar coordinates, \(r\) and \(\theta\), in place of \(x\) and \(y\) coordinates. Cylindrical coordinates are given in the order of \((r,\theta,z)\) and can be described as polar plus \(z\). The \(z\)-coordinate is measured the same way as in rectangular coordinates (signed distance above or below the \(xy\)-plane) and polar coordinates are measured as a projection of the point in three dimensions onto the \(xy\)-plane. To convert between rectangular and cylindrical coordinates, we use the same conversion equations as between rectangular and polar coordinates (in two dimensions).

• From cylindrical to rectangular

  Given cylindrical coordinates \((r,\theta,z)\) of a point \(P\), the rectangular coordinates \((x,y,z)\) of \(P\) satisfy \[\begin{aligned}x \amp= r \cos(\theta) \amp y \amp= r \sin(\theta) \amp z \amp= z\end{aligned}\].

• From rectangular to cylindrical

  Given rectangular coordinates \((x,y,z)\) of a point \(P\), the cylindrical coordinates \((r,\theta,z)\) of \(P\) satisfy \[\begin{aligned}r^2 \amp= x^2 + y^2 \amp \tan(\theta) \amp= \frac{y}{x} \amp z \amp= z\end{aligned}\] assuming \(x \neq 0\).

The activities that close this subsection are meant to give you not only some algebraic experience with the cylindrical coordinate measurements but also some examples where you can make sense of your results from a geometric perspective.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Cylindrical coordinates used an angular measurement in the \(xy\)-plane as part of the description of location. Spherical coordinates use one linear measurement and two angular measurements to specify the location of a point. The three measurements used to define the spherical coordinates of a point in \(\R^3\) are \(\rho\) (rho), \(\theta\), and \(\phi\) (phi), where

• \(\rho\)
• \(\theta\)
• \(\phi\)\(z\)

• From rectangular to spherical

  Given rectangular coordinates \((x,y,z)\) of a point \(P\), the spherical coordinates \((\rho,\theta,\phi)\) of \(P\) satisfy \[\begin{aligned}\rho \amp = \sqrt{x^2 + y^2 + z^2} \\ \tan(\theta) \amp = \frac{y}{x} \\ \cos(\phi) \amp = \frac{z}{\rho}\end{aligned}\] where in the second and third equations, we require \(x \ne 0\) and \(\rho \ne 0\).

• From spherical to rectangular

  Given spherical coordinates \((\rho,\theta,\phi)\) of a point \(P\), the rectangular coordinates \((x,y,z)\) of \(P\) satisfy \[\begin{aligned}x \amp = \rho \sin(\phi) \cos(\theta) \\ y \amp = \rho \sin(\phi) \sin(\theta) \\ z \amp = \rho \cos(\phi)\end{aligned}\].

Our first spherical coordinates activity focuses on understanding how spherical coordinates can be converted to rectangular coordinates and visualizing what the spherical coordinates measure.

Now that we have worked through how to think about points in spherical coordinates, we will do as we did with the other coordinate systems in this section and think about equations and surfaces.

The next activity gives you a chance to apply some of the skills introduced in the previous example.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.