7.2: Spherical Coordinates

Spherical coordinates are defined from Cartesian coordinates as

$$\begin{aligned} r &= \sqrt{x^2+y^2+z^2} \\[4pt] \phi &= \arctan(y/x) \\[4pt] \theta &=\arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right)\end{aligned} \nonumber$$

or alternatively

$$\begin{aligned} x &= r \cos\phi\sin\theta,\\[4pt] y &= r \sin\phi\sin\theta \\[4pt] z &=r \cos\theta\end{aligned} \nonumber$$

as indicated schematically in Fig. $\PageIndex{1}$.

Figure $\PageIndex{1}$: Spherical coordinates

Using the chain rule we find

$$\begin{aligned} \frac{\partial}{\partial x}{~} &= \frac{\partial r}{\partial x}\frac{\partial}{\partial r}{~} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi}{~} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}{~}\nonumber\\[4pt] &= \frac{x}{r} \frac{\partial}{\partial r}{~}-\frac{y}{x^2+y^2}\frac{\partial}{\partial \phi}{~} +\frac{xz}{r^2\sqrt{x^2+y^2}}\frac{\partial}{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\cos\phi\frac{\partial}{\partial r}{~}-\frac{\sin\phi}{r\sin\theta} \frac{\partial}{\partial \phi}{~} +\frac{\cos\phi\cos\theta}{r} \frac{\partial}{\partial \theta}{~},\\[4pt] \frac{\partial}{\partial y}{~} &= \frac{\partial r}{\partial y}\frac{\partial}{\partial r}{~}+\frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi}{~}+\frac{\partial \theta}{\partial y}\frac{\partial}{\partial \theta}{~}\nonumber\\[4pt] &= \frac{y}{r} \frac{\partial}{\partial r}{~}+\frac{x}{x^2+y^2}\frac{\partial}{\partial \phi}{~} +\frac{yz}{r^2\sqrt{x^2+y^2}}\frac{\partial}{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\sin\phi\frac{\partial}{\partial r}{~}+\frac{\cos\phi}{r\sin\theta} \frac{\partial}{\partial \phi}{~} +\frac{\sin\phi\cos\theta}{r} \frac{\partial}{\partial \theta}{~},\\[4pt] \frac{\partial}{\partial z}{~} &= \frac{\partial r}{\partial z}\frac{\partial}{\partial r}{~}+\frac{\partial \phi}{\partial z}\frac{\partial}{\partial \phi}{~}+\frac{\partial \theta}{\partial z}\frac{\partial }{\partial \theta}{~}\nonumber\\[4pt] &= \frac{z}{r} \frac{\partial }{\partial r}{~} -\frac{\sqrt{x^2+y^2}}{r^2}\frac{\partial }{\partial \theta}{~} \nonumber\\[4pt] &= \sin\theta\sin\phi\frac{\partial }{\partial r}{~}-\frac{\sin\theta}{r} \frac{\partial }{\partial \theta}{~}.\\[4pt]\end{aligned} \nonumber$$

once again we can write ${\nabla}$ in terms of these coordinates.

$$\begin{aligned} {\nabla} &=& \hat{e}_r \frac{\partial}{\partial r}{~}+\hat{e}_\phi \frac{1}{r\sin\theta}\frac{\partial}{\partial \phi}{~} + \hat{e}_\theta \frac{1}{r}\frac{\partial}{\partial \theta}{~}\end{aligned} \nonumber$$ where the unit vectors $$\begin{aligned} \hat{e}_r &=& (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta), \nonumber\\[4pt] \hat{e}_\phi &=& (-\sin\phi,\cos\phi,0), \nonumber\\[4pt] \hat{e}_\theta &=& (\cos\phi\cos\theta,\sin\phi\cos\theta,-\sin\theta).\end{aligned} \nonumber$$

are an orthonormal set. We say that spherical coordinates are orthogonal.

We can use this to evaluate $\Delta={\nabla}^2$,

$$\Delta = \frac{1}{r^2}\frac{\partial}{\partial r}{~}\left(r^2 \frac{\partial}{\partial r}{~}\right) +\frac{1}{r^2} \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}{~} \left( \sin\theta\frac{\partial}{\partial \theta}{~} \right) + \frac{1}{r^2}\frac{\partial^2}{\partial \phi^2}{~} \nonumber$$

Finally, for integration over these variables we need to know the volume of the small cuboid contained between $r$ and $r+\delta r$, $\theta$ and $\theta + \delta\theta$ and $\phi$ and $\phi+\delta\phi$.

Figure $\PageIndex{2}$: Integration in spherical coordinates

The length of the sides due to each of these changes is $\delta r$, $r \delta \theta$ and $r \sin \theta \delta \theta$, respectively (these are the Jacobians for the conversion of Cartesian coordinates to polar and spherical coordinates, respectively). We thus conclude that

$$\int_V f(x,y,z) dx dy dz = \int_V f(r,\theta,\phi) r^2\sin\theta \,dr \,d\theta \,d\phi. \nonumber$$

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