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 <math>r</math>, the angle measured in the x-y plane between the point and the x axis <math>\theta</math>, the distance perpendicular to the x-y plane: <math>(r, \theta. z)</math>

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

The equations converting the parameters are as follows:

<math>r^2 = x^2 + y^2</math>

<math>\tan{\theta} = \frac{y}{x}</math>

<math>x = r \cos{\theta}</math>

<math>y = r \sin{\theta}</math>

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