Cylindrical coordinates

Cylindrical coordinates refers to a three-dimensional coordinate system used to describe the location of a point in space based on the distance from the origin in the x-y plane $r$, the angle measured in the x-y plane between the point and the x axis $\theta$, the distance perpendicular to the x-y plane: $(r,\theta .z)$

In a sense, cylindrical coordinates are polar coordinates with a third dimension added: $(r,\theta )$ correspond to the polar coordinates for $(x,y)$. This is in contrast to spherical coordinates, where $z$ is replaced by an angle, just like x and y are in polar coordinates.

The equations converting the parameters are as follows:

$r^{2}=x^{2}+y^{2}$

$\tan {\theta }={\frac {y}{x}}$

$x=r\cos {\theta }$

$y=r\sin {\theta }$

In cylindrical coordinates, the Jacobian is $r$ so that ${\text{d}}x\,{\text{d}}y\,{\text{d}}z=r{\text{d}}r\,{\text{d}}\theta \,{\text{d}}z$.