path: root/source/know/concept/spherical-coordinates/index.md

---
title: "Spherical coordinates"
sort_title: "Spherical coordinates"
date: 2021-03-04
categories:
- Mathematics
- Physics
layout: "concept"
---

**Spherical coordinates** are an extension of polar coordinates $$(r, \varphi)$$ to 3D.
The position of a given point in space is described by
three variables $$(r, \theta, \varphi)$$, defined as:

*   $$r$$: the **radius** or **radial distance**: distance to the origin.
*   $$\theta$$: the **elevation**, **polar angle** or **colatitude**:
    angle to the positive $$z$$-axis, or **zenith**, i.e. the "north pole".
*   $$\varphi$$: the **azimuth**, **azimuthal angle** or **longitude**:
    angle from the positive $$x$$-axis, typically in the counter-clockwise sense.

Note that this is the standard notation among physicists,
but mathematicians often switch the definitions of $$\theta$$ and $$\varphi$$,
while still writing $$(r, \theta, \varphi)$$.

Cartesian coordinates $$(x, y, z)$$ and the spherical system
$$(r, \theta, \varphi)$$ are related by:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            x &= r \sin\theta \cos\varphi \\
            y &= r \sin\theta \sin\varphi \\
            z &= r \cos\theta
        \end{aligned}
    }
\end{aligned}$$

Conversely, a point given in $$(x, y, z)$$
can be converted to $$(r, \theta, \varphi)$$ using these formulae,
where $$\mathtt{atan2}$$ is the 2-argument arctangent,
which is needed to handle the signs correctly:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            r
            &= \sqrt{x^2 + y^2 + z^2}
            \\
            \theta
            &= \arccos(z / r)
            \\
            \varphi
            &= \mathtt{atan2}(y, x)
        \end{aligned}
    }
\end{aligned}$$

Spherical coordinates form
an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
whose **scale factors** $$h_r$$, $$h_\theta$$ and $$h_\varphi$$ we need.
To get those, we calculate the unnormalized local basis:

$$\begin{aligned}
    h_r \vu{e}_r
    &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
    \\
    &= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
    \\
    h_\theta \vu{e}_\theta
    &= \vu{e}_x \pdv{x}{\theta} + \vu{e}_y \pdv{y}{\theta} + \vu{e}_z \pdv{z}{\theta}
    \\
    &= \vu{e}_x \: r \cos{\theta} \cos{\varphi} + \vu{e}_y \: r \cos{\theta} \sin{\varphi} - \vu{e}_z \: r \sin{\theta}
    \\
    h_\varphi \vu{e}_\varphi
    &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
    \\
    &= - \vu{e}_x \: r \sin{\theta} \sin{\varphi} + \vu{e}_y \: r \sin{\theta} \cos{\varphi}
\end{aligned}$$

By normalizing the **local basis vectors**
$$\vu{e}_r$$, $$\vu{e}_\theta$$ and $$\vu{e}_\varphi$$,
we arrive at these expressions:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            h_r
            &= 1
            \\
            h_\theta
            &= r
            \\
            h_\varphi
            &= r \sin{\theta}
        \end{aligned}
    }
    \qquad\qquad
    \boxed{
        \begin{aligned}
            \vu{e}_r
            &= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
            \\
            \vu{e}_\theta
            &= \vu{e}_x \cos{\theta} \cos{\varphi} + \vu{e}_y \cos{\theta} \sin{\varphi} - \vu{e}_z \sin{\theta}
            \\
            \vu{e}_\varphi
            &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
        \end{aligned}
    }
\end{aligned}$$

Thanks to these scale factors, we can easily convert calculus from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.





## Differential elements

For line integrals,
the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows:

$$\begin{aligned}
    \boxed{
        \dd{\vb{\ell}}
        = \vu{e}_r \dd{r}
        + \: \vu{e}_\theta \: r \dd{\theta}
        + \: \vu{e}_\varphi \: r \sin{\theta} \dd{\varphi}
    }
\end{aligned}$$

For surface integrals,
the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:

$$\begin{aligned}
    \boxed{
        \dd{\vb{S}}
        = \vu{e}_r \: r^2 \sin{\theta} \dd{\theta} \dd{\varphi}
        + \: \vu{e}_\theta \: r \sin{\theta} \dd{r} \dd{\varphi}
        + \: \vu{e}_\varphi \: r \dd{r} \dd{\theta}
    }
\end{aligned}$$

And for volume integrals,
the infinitesimal volume $$\dd{V}$$ takes the following form:

$$\begin{aligned}
    \boxed{
        \dd{V}
        = r^2 \sin{\theta} \dd{r} \dd{\theta} \dd{\varphi}
    }
\end{aligned}$$



## Common operations

The basic vector operations (gradient, divergence, curl and Laplacian) are given by:

$$\begin{aligned}
    \boxed{
        \nabla f
        = \vu{e}_r \pdv{f}{r}
        + \vu{e}_\theta \frac{1}{r} \pdv{f}{\theta} + \mathbf{e}_\varphi \frac{1}{r \sin{\theta}} \pdv{f}{\varphi}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \nabla \cdot \vb{V}
        = \pdv{V_r}{r} + \frac{2}{r} V_r
        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{\cot{\theta}}{r} V_\theta
        + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \times \vb{V}
            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{\cot{\theta}}{r} V_\varphi
            - \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg)
            \\
            &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi}
            - \pdv{V_\varphi}{r} - \frac{V_\varphi}{r} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_\theta}{r} + \frac{V_\theta}{r}
            - \frac{1}{r} \pdv{V_r}{\theta} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \nabla^2 f
        = \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r}
        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{\cot{\theta}}{r^2} \pdv{f}{\theta}
        + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi}
    }
\end{aligned}$$



## Uncommon operations

Uncommon operations include:
the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$,
the gradient of a vector $$\nabla \vb{V}$$,
the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$,
the Laplacian of a vector $$\nabla^2 \vb{V}$$,
and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla (\nabla \cdot \vb{V})
            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r}
            + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
            \\
            &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
            + \frac{\cot{\theta}}{r} \pdv{V_\theta}{r} - \frac{2}{r^2} V_r - \frac{\cot{\theta}}{r^2} V_\theta \bigg)
            \\
            &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta}
            \\
            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
            - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
            \\
            &\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \vb{V}
            &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r} + \vu{e}_r \vu{e}_\theta \pdv{V_\theta}{r} + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
            \\
            &\quad\: + \vu{e}_\theta \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\theta} - \frac{V_\theta}{r} \bigg)
            + \vu{e}_\theta \vu{e}_\theta \bigg( \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_r}{r} \bigg)
            + \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta}
            \\
            &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{\cot{\theta}}{r} V_\varphi \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \vu{e}_\varphi
            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{\cot{\theta}}{r} V_\theta \bigg)
        \end{aligned}
        }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            (\vb{U} \cdot \nabla) \vb{V}
            &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\theta}{r} \pdv{V_r}{\theta}
            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg)
            \\
            &\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta}
            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{\cot{\theta}}{r} U_\varphi V_\varphi \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta}
            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{\cot{\theta}}{r} U_\varphi V_\theta \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla^2 \vb{V}
            &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
            + \frac{2}{r} \pdv{V_r}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_r}{\theta}
            \\
            &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
            - \frac{2}{r^2} V_r - \frac{2 \cot{\theta}}{r^2} V_\theta \bigg)
            \\
            &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r}
            \\
            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
            - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi}
            \\
            &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_\varphi}{\theta}
            - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \cdot \overline{\overline{\mathbf{T}}}
            &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi}
            \\
            &\qquad\qquad + \frac{2}{r} T_{rr} + \frac{\cot{\theta}}{r} T_{\theta r} - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
            \\
            &\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta}
            + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi}
            \\
            &\qquad\qquad + \frac{2}{r} T_{r \theta} + \frac{T_{\theta r}}{r}
            + \frac{\cot{\theta}}{r} T_{\theta \theta} - \frac{\cot{\theta}}{r} T_{\varphi \varphi} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta}
            + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi}
            \\
            &\qquad\qquad + \frac{2}{r} T_{r \varphi} + \frac{\cot{\theta}}{r} T_{\theta \varphi}
            + \frac{T_{\varphi r}}{r} + \frac{\cot{\theta}}{r} T_{\varphi \theta} \bigg)
        \end{aligned}
    }
\end{aligned}$$



## References
1.  M.L. Boas,
    *Mathematical methods in the physical sciences*, 2nd edition,
    Wiley.
2.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.