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 $$x \ = \ r \cos \theta,\qquad y \ = \ r \sin \theta, \qquad z \ = \ z\,.$$

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 $\phi$?