M344: Calculus III
Section 15.8
Spherical Coordinates¶
We begin by defining spherical coordinates in $\mathbb{R}^3$.
Video Lecture, part I
Video Lecture, part II
The Spherical Transformation¶
As mentioned in the videos above, we need to set some conventions so that we're all computing in the same way. We'll reserve the right to change these conventions in the future, if there's a good reason. But keep in mind that changing these conventions could (and probably will) change the Jacobian and volume element as well.
In order to guarantee that our coordinates are unique, we've assumed that $\rho \geq 0$ measures the radial length in the direction of the positive $z$-axis; $0 \leq \varphi \leq \pi$ is the angle in the $xz$-plane, measured from the positive $z$-axis in the direction of the positive $x$-axis; and $0 \leq \theta < 2\pi$ measures the angle in the $xy$-plane in the standard way: from the positive $x$-axis in the direction of the positive $y$-axis.
The transformation that takes spherical coordinates to Cartesian coordinates is then given by
$$
T(\rho,\varphi,\theta) = \begin{cases}
x = \rho \sin\varphi\cos\theta,\\[0.5 ex]
y = \rho \sin\varphi\sin\theta,\\[0.5 ex]
z = \rho \cos\varphi.
\end{cases}
$$
Compute the Jacobian of this transformation, and use it to write down a formula for $dV$ in spherical coordinates.
Check your answer
The Jacobian is $$ \left\vert \frac{\partial(x,y,z)}{\partial(\rho,\varphi,\theta)} \right\vert = \rho^2\sin\varphi. $$
Therefore, the spherical volume element is $$ dV = \rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta. $$
Video Solution
Since the book has not yet talked about transformations and Jacobians in section 15.8, they provide a purely geometric argument to compute the volume of a small "spherical box." The interested reader should read this argument in the book. However, if you are comfortable with the transformation lecture and the idea of the Jacobian, then you will not miss any important ideas by skipping that derivation in the book.
1. Example¶
A solid lies above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere $x^2 + y^2 + z^2 = z$. Describe the solid in spherical coordinates, then compute its volume.
Check your answer
This intersection of this solid with the $yz$-plane is:
The solid region obtained by rotating the shaded region about the $z$-axis. Its volume is$$ \frac{\pi}{8} $$
Video Solution
2. Example¶
Find the average distance from a point inside a ball of radius $a$ to its center.
Check your answer
$$\frac{3a}{4}$$
Video Solution
3. Example¶
Find the volume of the smaller wedge cut from a sphere of radius $a$ by two planes that intersect along a diameter at an angle of $\frac{\pi}{6}$.
Check your answer
$$\frac{\pi a^3}{9}$$
Video Solution
Discussion¶
Questions? You can ask them here.