| Version current |
Version 8 |
| \PMlinkescapeword{bottom} |
\PMlinkescapeword{bottom} |
| \PMlinkescapeword{polar} |
\PMlinkescapeword{polar} |
| \PMlinkescapeword{top} |
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| \emph{Spherical coordinates} are a system of coordinates for $\R^3$, |
\emph{Spherical coordinates} are a system of coordinates for $\R^3$, |
| or more generally $\R^n$. |
or more generally $\R^n$. |
| One coordinate is the distance from the origin, |
One coordinate is the distance from the origin, |
| which can be thought of as |
which can be thought of as |
| the radius of the sphere centred at the origin on which the point lies. |
the radius of the sphere centred at the origin on which the point lies. |
| The other coordinates are angles that specify the position of the point on this sphere. |
The other coordinates are angles that specify the position of the point on this sphere. |
|
|
| In $\R^3$ the coordinates are given by |
In $\R^3$ the coordinates are given by |
| \begin{align*} |
\begin{align*} |
| \left( |
\left( |
| \begin{tabular}{c} |
\begin{tabular}{c} |
| $x$\\ |
$x$\\ |
| $y$\\ |
$y$\\ |
| $z$ |
$z$ |
| \end{tabular}\right)&= |
\end{tabular}\right)&= |
| \left( |
\left( |
| \begin{tabular}{c} |
\begin{tabular}{c} |
| $r\sin\phi\cos\theta$\\ |
$r\sin\phi\cos\theta$\\ |
| $r\sin\phi\sin\theta$\\ |
$r\sin\phi\sin\theta$\\ |
| $r\cos\phi$ |
$r\cos\phi$ |
| \end{tabular} |
\end{tabular} |
| \right), |
\right), |
| \end{align*} |
\end{align*} |
| where $r$ is the distance from the origin, |
where $r$ is the distance from the origin, $\theta$ is the azimuthal angle defined for $\theta\in[0,2\pi)$, and $\phi\in[0,\pi]$ is the polar angle. |
| $\theta$ is the \emph{azimuthal angle} defined for $\theta\in[0,2\pi)$, |
|
| and $\phi\in[0,\pi]$ is the \emph{polar angle}. |
|
| Note that $\phi=0$ corresponds to the top of the sphere and $\phi=\pi$ corresponds to the bottom of the sphere. |
Note that $\phi=0$ corresponds to the top of the sphere and $\phi=\pi$ corresponds to the bottom of the sphere. |
| There is a clash between the mathematicians' and the physicists' definition of spherical coordinates, interchanging both the direction of $\phi$ and the choice of names for the two angles (physicists often use $\theta$ as the azimuthal angle and $\phi$ as the polar one). |
There is a clash between the mathematicians' and the physicists' definition of spherical coordinates, interchanging both the direction of $\phi$ and the choice of names for the two angles (physicists often use $\theta$ as the azimuthal angle and $\phi$ as the polar one). |
|
|
| Spherical coordinates are a generalization of polar coordinates, |
Spherical coordinates are a generalization of polar coordinates, |
| and can be further generalized to $\R^n$, |
and can be further generalized to $\R^n$, |
| with $n-2$ polar angles $\phi_1,\ldots,\phi_{n-2}$ and one azimuthal angle $\theta$: |
with $n-2$ polar angles $\phi_1,\ldots,\phi_{n-2}$ and one azimuthal angle $\theta$: |
| \begin{align*} |
\begin{align*} |
| \left( |
\left( |
| \begin{tabular}{c} |
\begin{tabular}{c} |
| $x_1$\\ |
$x_1$\\ |
| $x_2$\\ |
$x_2$\\ |
| $\vdots$\\ |
$\vdots$\\ |
| $x_k$\\ |
$x_k$\\ |
| $\vdots$\\ |
$\vdots$\\ |
| $x_{n-1}$\\ |
$x_{n-1}$\\ |
| $x_n$ |
$x_n$ |
| \end{tabular}\right)&= |
\end{tabular}\right)&= |
| \left( |
\left( |
| \begin{tabular}{c} |
\begin{tabular}{c} |
| $r\cos\phi_1$\\ |
$r\cos\phi_1$\\ |
| $r\sin\phi_1\cos\phi_2$\\ |
$r\sin\phi_1\cos\phi_2$\\ |
| $\vdots$\\ |
$\vdots$\\ |
| $r\left(\prod_{i=1}^{k-1}\sin\phi_i\right)\cos\phi_k$\\ |
$r\left(\prod_{i=1}^{k-1}\sin\phi_i\right)\cos\phi_k$\\ |
| $\vdots$\\ |
$\vdots$\\ |
| $r\sin\phi_1\sin\phi_2\cdots\cos\theta$\\ |
$r\sin\phi_1\sin\phi_2\cdots\cos\theta$\\ |
| $r\sin\phi_1\sin\phi_2\cdots\sin\phi_{n-2}\sin\theta$. |
$r\sin\phi_1\sin\phi_2\cdots\sin\phi_{n-2}\sin\theta$. |
| \end{tabular} |
\end{tabular} |
| \right). |
\right). |
| \end{align*} |
\end{align*} |
|
|
| These are sometimes called \emph{hyperspherical coordinates} if $n>3$. |
These are sometimes called \emph{hyperspherical coordinates} if $n>3$. |
