![[spherical_coordinates.png]] Spherical coordinates in the physics convention $(r, \theta, \phi)$ are described via the figure above. The angle $\phi$ is the azimuthal angle in the $(x, y)$ plane and the angle $\theta$ is the polar angle with respect to the $z$-axis. The radius is denoted by $r$. We therefore have for each point $(x, y, z)\in\mathbb{R}$ that $ \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}r\sin\theta\cos\phi\\ r\sin\theta\sin\phi\\ r\cos\theta\end{bmatrix}. $ The Jacobian of this transformation is given by $ J:=\frac{\partial(x, y, z)}{\partial (r, \theta, \phi)} = \begin{bmatrix}\sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi\\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi\\ \cos\theta & -r\sin\theta & 0 \end{bmatrix}. $ We hence have that $ \begin{bmatrix} \frac{\partial }{\partial r}\\ \frac{\partial}{\partial \theta}\\ \frac{\partial}{\partial\phi} \end{bmatrix} = \begin{bmatrix}\sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi\\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi\\ \cos\theta & -r\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{bmatrix}. $ The inverse transformation is given by $ \begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ r^{-1}\cos\theta\cos\phi & r^{-1}\cos\theta\sin\phi & -r^{-1}\sin\theta\\ -r^{-1}\frac{\sin\phi}{\sin\theta} & r^{-1}\frac{\cos\phi}{\sin\theta} & 0\\ \end{bmatrix} \begin{bmatrix} \frac{\partial }{\partial r}\\ \frac{\partial}{\partial \theta}\\ \frac{\partial}{\partial\phi} \end{bmatrix}. $ Furthermore, we have that $\det J = r^2\sin\theta$. For the Laplacian of a function $u$ we obtain the formula $ \nabla^2 u = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial u}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial u}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 u}{\partial\phi^2}. $ ## References - [Spherical Coordinate System on Wikipedia](https://en.wikipedia.org/wiki/Spherical_coordinate_system)