Spherical coordinates

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We integrate over regions in spherical coordinates.

Another way to generalize polar coordinates to three dimensions is with spherical coordinates.

An ordered triple consisting of a radius, an angle, and a height $(\rho ,\theta , \varphi )$ can be graphed as $\begin{align*} x &= \rho\cdot \cos(\theta)\sin(\varphi)\\ y &= \rho\cdot \sin(\theta)\sin(\varphi)\\ z &= \rho\cdot \cos(\varphi) \end{align*}$

meaning:

Coordinates of this type are called spherical coordinates.

Consider the point $(\rho ,\theta ,\phi )=(2,-\pi /4,\pi /4)$ in spherical coordinates. What is this point when expressed in $(x,y,z)$ -coordinates? $(x,y,z) = \left (\answer {2\cos (-\pi /4)\sin (\pi /4)}, \answer {2 \sin (-\pi /4)\sin (\pi /4)},\answer {2\cos (\pi /4)}\right )$

Triple integrals in spherical coordinates

If you want to evaluate this integral $\iiint _R F \d V,$ you have to change $R$ to a region defined in $(x,y,z)$ -coordinates, and change $\d V$ to some combination of $\d x\d y\d z$ leaving you with some iterated integral: $\int _a^b\int _c^d\int _p^q F(x,y,z) \d y \d x\d z$ Now consider representing a region $R$ in spherical coordinates and let’s express $\d V$ in terms of $\d \rho$ , $\d \phi$ , and $\d \theta$ . To do this, consider the diagram below:

Here we see $\begin{align*} \d V &= (\rho\sin(\varphi)\d\theta)\cdot(\rho \d \varphi)\cdot (\d \rho)\\ &= \rho^2 \sin(\varphi)\d\rho\d\varphi\d\theta. \end{align*}$

Recalling that the determinate of a $3\times 3$ matrix gives the volume of a parallelepiped, we could also deduce the correct form for $\d V$ by setting $\begin{align*} x(\rho,\theta,\phi) &= \rho \cos(\theta)\sin(\phi)\\ y(\rho,\theta,\phi) &= \rho \sin(\theta)\sin(\phi)\\ z(\rho,\theta,\phi) &= \rho \cos(\phi) \end{align*}$

and computing:

$\begin{align*} \d V &= \left| \det \begin{bmatrix} \pp[x]{\rho} \d \rho & \pp[y]{\rho} \d \rho & \pp[z]{\rho} \d \rho\\ \pp[x]{\theta} \d \theta & \pp[y]{\theta} \d \theta & \pp[z]{\theta} \d \theta\\ \pp[x]{\phi} \d \phi & \pp[y]{\phi} \d \phi & \pp[z]{\phi} \d \phi \end{bmatrix} \right|\\ &= \left| \det \begin{bmatrix} \answer[given]{\cos(\theta)\sin(\phi)} \d \rho & \answer[given]{\sin(\theta)\sin(\phi)} \d \rho & \answer[given]{\cos(\phi)} \d \rho \\ \answer[given]{-\rho\sin(\theta)\sin(\phi)} \d \theta & \answer[given]{\rho\cos(\theta)\sin(\phi)} \d \theta & 0\\ \answer[given]{\rho\cos(\theta)\cos(\phi)} \d \phi & \answer[given]{\rho\sin(\theta)\cos(\phi)} \d \phi & \answer[given]{-\rho\sin(\phi)}\d \phi \end{bmatrix} \right|\\ &= \rho^2 \sin(\varphi)\d\rho\d\varphi\d\theta \end{align*}$

We may now state at theorem:

Fubini Let $F:\R ^3\to \R$ be continuous on the region $R=\{(\rho ,\theta ,\phi ):\text {$\alpha \leq \theta \leq \beta $, $a\leq \phi \leq b$, $G_1(\theta ,\phi )\le \rho \le G_2(\theta ,\phi )$}\}$ Then: $\begin{align*} \iiint_R &F(\rho,\theta,\phi)\d V\\ &= \int_\alpha^\beta\int_{a}^{b} \int_{G_1(\theta,\phi)}^{G_2(\theta,\phi)}F(\rho,\theta,\phi) \rho^2 \sin(\phi)\d\rho\d\varphi\d\theta. \end{align*}$

Write down a triple integral in spherical coordinates that will compute the volume of a sphere of radius $a$ . $\iiint _R \d V = \int _{\answer {0}}^{\answer {2\pi }} \int _{\answer {0}}^{\answer {\pi }} \int _{\answer {0}}^{\answer {a}} \rho ^2 \sin (\varphi )\d \rho \d \varphi \d \theta$