cylindrical coordinatesCylindricalCoordinates2007-05-02 02:00:162007-05-07 11:53:02Definition\usepackage{amssymb}
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\emph{Cylindrical coordinates} are a system of coordinates for $\mathbb{R}^3$. Two of the coordinates correspond to the polar coordinates of $\mathbb{R}^2$, and the third coordinate corresponds with the $z$ axis. Thus, the coordinates are given by
$$\left( \begin{array}{c}
x \\
y \\
z \end{array} \right)=\left( \begin{array}{c}
r \cos \theta \\
r \sin \theta \\
z \end{array} \right),$$
where $r$ is the distance from $(0,0,0)$ to $(x,y,0)$ and $\theta$ is the azimuthal angle defined for $\theta \in [0,2\pi )$.
Just as with polar coordinates, one can convert from Cartesian coordinates to cylindrical coordinates for any point not lying on the $z$ axis via
\begin{eqnarray*}
r(x,y) &=& \sqrt{x^2+ y^2}, \\
\theta(x,y) &=& \arctan (x,y),
\end{eqnarray*}
where $\arctan$ is defined \PMlinkname{here}{OperatornamearcTanWithTwoArguments}.