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Cylindrical and spherical coordinates

Cylindrical and Spherical Coordinates

The change-of-variables formula with 3 (or more) variables is just like the formula for two variables. If we do a change-of-variables $\Phi$ from coordinates $(u,v,w)$ to coordinates $(x,y,z)$, then the Jacobian is the determinant $$\frac{\partial(x,y,z)}{\partial(u,v,w)} \ = \ \left | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{matrix} \right |,$$ and the volume element is $$dV \ = \ dx\,dy\,dz \ = \ \left | \frac{\partial(x,y,z)}{\partial(u,v,w)}\right | du\,dv\,dw.$$

After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called cylindrical polar coordinates) and spherical coordinates (sometimes called spherical polar coordinates).

Cylindrical Coordinates: When there's symmetry about an axis, it's convenient to take the $z$-axis as the axis of symmetry and use polar coordinates $(r,\, \theta)$ in the $xy$-plane to measure rotation around the $z$-axis. Check the interactive figure to the right. A point $P$ is specified by coordinates $(r,\, \theta, \, z)$ where $z$ is the height of $P$ above the $xy$-plane.

(i) What happens to $P$ as $z$ changes?

(ii) What's the relation between $r$, $P$ and the axis of symmetry?

(iii) What are the natural restrictions on $\theta$?

(iv) The relation between Cartesian coordinates $(x,\,y,\,z)$ and Cylindrical coordinates $(r,\, \theta, \, z)$ for each point $P$ in $3$-space is $$ x \ = \ r \cos \theta,\qquad y \ = \ r \sin \theta, \qquad z \ = \ z\,.$$

Problem: Find the Jacobian of the transformation $(r,\theta,z) \to (x,y,z)$ of cylindrical coordinates.

Solution: This calculation is almost identical to finding the Jacobian for polar coordinates. Our partial derivatives are: \begin{eqnarray*} \frac{\partial x}{\partial r} = \cos(\theta), & \frac{\partial x}{\partial \theta} = -r \sin(\theta), & \frac{\partial x}{\partial z} = 0, \\ \frac{\partial y}{\partial r} = \sin(\theta), & \frac{\partial y}{\partial \theta} = r \cos(\theta), & \frac{\partial y}{\partial z} = 0, \cr \frac{\partial z}{\partial r} = 0, & \frac{\partial z}{\partial \theta} = 0, & \frac{\partial z}{\partial z} = 1. \cr \end{eqnarray*}

Our Jacobian is then the $3\times 3$ determinant $$\frac{\partial(x,y,z)}{\partial(r,\theta,z)} \ = \ \left | \begin{matrix} \cos(\theta) & -r \sin(\theta) & 0 \cr \sin(\theta) & r \cos(\theta) & 0 \cr 0&0&1 \end{matrix} \right | \ = \ r,$$ and our volume element is $dV = dx\, dy\,dz = r \, dr\,d\theta\,dz$.

Spherical Coordinates: A sphere is symmetric in all directions about its center, so it's convenient to take the center of the sphere as the origin. Then we let $\rho$ be the distance from the origin to $P$ and $\phi$ the angle this line from the origin to $P$ makes with the $z$-axis. Finally, as before, we use $\theta$ from polar coordinates in the $xy$-plane to measure rotation around the $z$-axis. Investigate the interactive figure to the right. A point $P$ is specified by $3$ coordinates $(\rho,\, \theta, \, \phi)$.
[Warning: Most physics texts swap the roles of $\theta$ and $\phi$.]
(i) The relation between Cartesian coordinates $(x,\,y,\,z)$ and Spherical Polar coordinates $(\rho,\, \theta, \, \phi)$ for each point $P$ in $3$-space is $$\ x \ = \ \rho \cos \theta \sin \phi,\qquad y \ = \ \rho \sin \theta \sin \phi, \qquad z \ = \ \rho \cos \phi\,.$$ (ii) The natural restrictions on $\rho,\, \theta,$ and $\phi$ are $$\color{rgb(0, 51, 153)}{ 0\, \le \, \rho \, < \, \infty\,, \qquad 0\, \le \, \theta \, < \, 2\pi \,, \qquad 0\, \le \, \phi \, \le \, \pi\,.}$$ (iii) Points on the earth are frequently specified by Latitude and Longitude. How do these relate to $\theta$ and $\phi$?

Problem: Find the Jacobian of the transformation $(\rho,\theta, \phi) \to (x,y,z)$ of spherical coordinates.

Solution: Now our partial derivatives are: \begin{eqnarray*} \frac{\partial x}{\partial \rho} = \cos(\theta)\sin(\phi), & \frac{\partial x}{\partial \theta} = -\rho \sin(\theta) \sin(\phi), & \frac{\partial x}{\partial \phi} = \rho \cos(\theta)\cos(\phi), \\ \frac{\partial y}{\partial \rho} = \sin(\theta)\sin(\phi), & \frac{\partial y}{\partial \theta} = \rho \cos(\theta) \sin(\phi), & \frac{\partial y}{\partial \phi} = \rho \sin(\theta)\cos(\phi), \\ \frac{\partial z}{\partial \rho} = \cos(\phi), & \frac{\partial z}{\partial \theta} = 0, & \frac{\partial z}{\partial \phi} = -\rho \sin(\phi). \end{eqnarray*}

Our Jacobian $\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}$ is then the $3\times 3$ determinant $$\quad \left | \begin{matrix} \cos(\theta)\sin(\phi) & -\rho \sin(\theta)\sin(\phi) & \rho \cos(\theta) \cos(\phi) \cr \sin(\theta)\sin(\phi) & \rho \cos(\theta)\sin(\phi) & \rho \sin(\theta) \cos(\phi) \cr \cos(\phi) & 0 & - \rho \sin(\phi). \end{matrix} \right |\quad$$ which works out to $\rho^2 \sin(\phi)$, and our volume element is $dV = dx\, dy\,dz = \rho^2 \sin(\phi) \, d\rho\, d\theta\,d\phi$.

Problem: Compute the volume of the ball $\rho \le R$ or radius $R$.

Solution: If $B$ is the unit ball, then its volume is $\iiint_B 1 dV$. We convert to spherical coordinates to get

\begin{eqnarray*} \hbox{Vol}(B) & = & \int_0^{\pi}\int_0^{2\pi} \int_0^R \rho^2 \sin(\phi) d\rho d\theta d\phi \\ & = & \int_0^\pi \int_0^{2\pi} \frac{R^3\sin(\phi)}{3} d\theta d\phi \\ & = & \int_0^\pi \frac{2 \pi R^3 \sin(\phi)}{3} d\phi \\ & = & \frac{4 \pi R^3}{3}. \end{eqnarray*}

Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Chapter 15: Multiple Integrals

Learning module LM 15.1: Multiple integrals

Learning module LM 15.2: Multiple integrals over rectangles:

Learning module LM 15.3: Double integrals over general regions:

Learning module LM 15.4: Double integrals in polar coordinates:

Learning module LM 15.5a: Multiple integrals in physics:

Learning module LM 15.5b: Integrals in probability and statistics:

Learning module LM 15.10: Change of variables:

Cylindrical and spherical coordinates