Lecture 17 Spherical Coordinates

Text References: Course notes pp. 66-74 & Rogawski 15.3-15.6

17.1 Recap

Last time, we discussed changing variables to cylindrical coordinates.

Exercise 17.1 Which of the following integrals is equal to \(\displaystyle \int_{-5}^5 \int_0^3\int_{-\sqrt{25-x^2}}^{\sqrt{25-x^2}} x~dydzdx\)?

  1. \(\displaystyle \int_{0}^3 \int_0^3\int_{0}^{\pi} \rho^2\cos(\phi)~d\phi dzd\rho\)
  2. \(\displaystyle \int_{0}^5 \int_0^3\int_{0}^{\pi} \rho^2\cos(\phi)~d\phi dzd\rho\)
  3. \(\displaystyle \int_{0}^3 \int_0^5\int_{0}^{2\pi} \rho\cos(\phi)~d\phi dzd\rho\)
  4. \(\displaystyle \int_{0}^5 \int_0^3\int_{0}^{2\pi} \rho^2\cos(\phi)~d\phi dzd\rho\)

Solution. The region of integration is a cylinder of radius \(\rho=5\) and height \(z=3\). The integrand \(x~dydzdx\) becomes \(\rho\cos(\phi)\cdot \rho = \rho^2\cos(\phi)\). The corresponding integral is d.

17.2 Learning Objectives

  • Evaluate triple integrals by using an appropriate change to spherical coordinates.

17.3 Spherical Coordinates

Spherical coordinates are another way to generalize polar coordinates using the distance \(r\) from the origin and two angles. The angle \(\phi\) is defined in the same way as it is for cylindrical coordinates (i.e. \(0\leq \phi < 2\pi\)). The angle \(\theta\) is the angle away from the positive \(z\)-axis (i.e. \(0\leq \theta \leq \pi\)). The transformation is given by \[x=r\sin(\theta)\cos(\phi), \quad y=r\sin(\theta)\sin(\phi), \quad \mbox{and}\quad z= r\cos(\theta)\]

This interactive applet might give you a geometric intuition for this coordinate system.

The Jacobian of this change of variables is \[\frac{\partial (x,y,z)}{\partial (r,\theta, \phi)}=\det \begin{bmatrix}x_{r}& x_{\theta}& x_{\phi} \\ y_{r}& y_{\theta}& y_{\phi} \\ z_{r}& z_{\theta}& z_{\phi} \end{bmatrix}=r^2\sin(\theta)\]

So, when changing from Cartesian to spherical coordinates, we can replace \(dV\) with \(r^2\sin(\theta)~drd\theta d\phi\).

Exercise 17.2 Evaluate \(\displaystyle \iiint_D \frac{1}{x^2+y^2+z^2}dV\) where \(D\) is the spherical shell between the spheres of radius \(3\) and radius \(5\) centred at the origin.

Solution. Note that a sphere of radius \(k\) can be described in spherical coordinates by \(0\leq r\leq k\), \(0\leq \theta \leq \pi\), and \(0\leq \phi < 2\pi\). Using this information helps us describe the region of integration as follows: \[3\leq r\leq 5, \quad \leq \theta \leq \pi, \quad \mbox{and} \quad 0\leq \phi < 2\pi\]

We can therefore describe the wedge using the following inequalities: \[0\leq \rho\leq 2, \quad 0\leq \phi\leq \pi,\quad \mbox{and}\quad 0\leq z \leq 3\rho\sin(\phi)\]

The integrand \(\dfrac{1}{x^2+y^2+z^2}\) becomes \(\dfrac{1}{r}\) in spherical coordinates. Putting everything together and applying the change of variables, we get

\[\begin{align*} \iiint_D \frac{1}{x^2+y^2+z^2}dV & = \int_0^{2\pi}\int_0^{\pi}\int_3^5 \frac{1}{r^2}r^2\sin(\theta)~dr d\theta d\phi \\ & = \int_0^{2\pi}\int_0^{\pi} 2\sin(\theta) d\theta d\phi \\ &= \int_0^{2\pi} 4 d\phi\\ & = 8\pi \end{align*}\]

17.4 Spherical vs Cylindrical Coordinates

At this point, you might be wondering how to decide which coordinate system is best suited for a particular problem. Here is a general rule of thumb:

  • Use cylindrical coordinates when there is symmetry about one of the three axes
  • Use spherical coordinates when there is symmetry about the origin