This page covers the basics of spherical coordinates and reference equations that use spherical coordinates for multi-variable calculus.
Spherical coordinates are very different from rectangular and cylindrical coordinates. There are still three coordinates, usually labeled \((\rho, \theta, \phi)\), but they are assigned so that spherical-type objects are easy to express. All distances and angles are measured based on a radial line drawn from the origin to the point. Table 1 describes how they are defined and Figure 1 shows this graphically.
\(\rho\) is the radial distance out from the origin to the point
\(\theta\) is the angle measured in the xy-plane from the positive x-axis to the shadow of the radial line. Note: This is the same angle as \(\theta\) in cylindrical coordinates.
\(\phi\) is the angle measured from the positive z-axis to the radial line
Note - We have specifically chosen \((\rho, \theta, \phi)\) to describe spherical coordinates but not all books, instructors and videos use these same variables. Some use \((r,\theta,\phi)\) with \(\theta\) and \(\phi\) meaning different things depending on the context. (See the Wiktionary page on spherical coordinates for examples.) Check with your instructor and textbook to see which one they require.
\((x,y,z) \to (\rho, \theta, \phi)\)
The equations to convert from rectangular to spherical coordinates, in Table 2, are somewhat complicated. So take a few minutes to learn and memorize them (why memorize?). Look for similarities and differences between the equations and see if you can make some sense out of them.
Here is a quick video clip discussing these equations and showing a neat animation that will help you get a feel for spherical coordinates.
Converting between Rectangular and Spherical Coordinates
Rectangular → Spherical
In order to convert from rectangular to spherical coordinates, the equations are Table 2 above are directly applied. The technique is not hard but it will help to watch a couple of quick videos showing how to do this with examples. The first video explains how to convert points from rectangular to spherical. The second video explains how to convert equations from rectangular to spherical. Both videos have plenty of examples.
To be able to use spherical coordinates in triple integrals, you need to understand the unit vectors in spherical coordinates. Here is a video that will help.
Before we go on, let's do a few practice problems.
Find an equation in spherical coordinates for the rectangular equation \( x^2 + y^2 - z^2 = 0 \).
Spherical → Rectangular
Sometimes it will be easier to evaluate an integral in rectangular coordinates rather than spherical coordinates. In this case, you may be given an equation in spherical coordinates and be asked to convert to rectangular coordinates.
To do this conversion, there is no set of equations with specific techniques. You just have to use trig identities and intuition to get an equation in rectangular coordinates. However, it sometimes helps to convert tangent, cotangent, secant and cosecant to sines and cosines and use \(\cos^2(t) + \sin^2(t) = 1\). Try these practice problems to get the idea.
Instructions - Unless otherwise instructed, convert these equations in spherical coordinates to rectangular coordinates.
\( \rho = 2 \csc(\phi) \)
\( \rho = 2 \csc(\phi) \)
\( x^2 + y^2 = 4 \)
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