This page covers the basics of spherical coordinates and reference equations that use spherical coordinates for multivariable calculus.
Recommended Books on Amazon (affiliate links)  

Join Amazon Student  FREE TwoDay Shipping for College Students 
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 sphericaltype 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 xyplane from the positive xaxis 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 zaxis to the radial line 
Table 1 

Figure 1 

[source: Pauls Online Notes] 
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.
\((x,y,z) \to (\rho, \theta, \phi)\) 

\(x=\rho\sin\phi\cos\theta\) 
\(y=\rho\sin\phi\sin\theta\) 
\(z=\rho\cos\phi\) 
\(\rho^2=x^2+y^2+z^2\) 
Table 2 
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.
video by MIP4U 

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.
video by Krista King Math 

video by MIP4U 

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.
video by Michel vanBiezen 

Before we go on, let's do a few practice problems.
Practice
Find an equation in spherical coordinates for the rectangular equation \( x^2 + y^2  z^2 = 0 \).
Problem Statement 

Find an equation in spherical coordinates for the rectangular equation \( x^2 + y^2  z^2 = 0 \).
Solution 

close solution

Log in to rate this practice problem and to see it's current rating. 

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.
Practice
Unless otherwise instructed, convert these equations in spherical coordinates to rectangular coordinates.
\( \rho = 2 \csc(\phi) \)
Problem Statement 

Convert \( \rho = 2 \csc(\phi) \) to rectangular coordinates.
Final Answer 

\( x^2 + y^2 = 4 \)
Problem Statement 

Convert \( \rho = 2 \csc(\phi) \) to rectangular coordinates.
Solution 

Final Answer 

\( x^2 + y^2 = 4 \)
close solution

Log in to rate this practice problem and to see it's current rating. 

You CAN Ace Calculus
external links you may find helpful 

Wikipedia  Connection between polar coordinates and spherical and cylindrical coordinates 
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1  basic identities  

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) 
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) 
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) 
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) 
Set 2  squared identities  

\( \sin^2t + \cos^2t = 1\) 
\( 1 + \tan^2t = \sec^2t\) 
\( 1 + \cot^2t = \csc^2t\) 
Set 3  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle formulas  

\(\displaystyle{ \sin^2(t) = \frac{1\cos(2t)}{2} }\) 
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) 
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\) 
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
Trig Integrals
\(\int{\sin(x)~dx} = \cos(x)+C\) 
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\) 
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) 
\(\int{\csc(x)~dx} = \) \( \ln\abs{\csc(x)+\cot(x)}+C\) 
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. 