Cylindrical Coordinates#

Cylindrical coordinates are a natural extension of the Polar Coordinates to three dimensions. They identify each point in the Euclidean space \(\mathbb{R}^3\) by its distance from the \(z\)-axis, the angle \(\phi\) its projection in the \(xy\)-plane makes with the \(x\)-axis and its \(z\) component.

Definition: Radial Distance

The number \(\rho\) is known as the radial distance.

Warning

This is not the same as the radial distance used in spherical coordinates.

Definition: Azimuthal Angle (Azimuth)

The number \(\phi\) is known as the azimuthal angle or azimuth.

Definition: Axial Coordinate (Height)

The number \(z\) is known as the axial coordinate or height.

Note

Despite the word "height", the axial coordinate's algebraic sign is crucial.

Theorem: Local Coordinate Basis of Cylindrical Coordinates

Let \(\tau: (0, +\infty) \times (0, 2\uppi) \times \mathbb{R} \to \mathbb{R}^3\) be the transition map from cylindrical coordinates to Cartesian coordinates.

Its normalized local coordinate basis at \((\rho, \varphi, z)\) is the following:

\[\boldsymbol{\hat{\rho}} = \begin{bmatrix}\cos \varphi \\ \sin \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{\hat{z}} = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\]

Proof

We have the following by definition:

\[\tau\left(\begin{bmatrix}\rho \\ \varphi \\ z \end{bmatrix}\right) = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \\ z \end{bmatrix}\]

We see that \(\tau\) is totally differentiable on \((0, +\infty) \times (0, 2\uppi) \times \mathbb{R}\) with the following Jacobian matrix:

\[J_{\tau}\left(\begin{bmatrix}\rho \\ \varphi \\ z \end{bmatrix}\right) = \begin{bmatrix}\cos \varphi & -\rho \sin \varphi & 0 \\ \sin \varphi & \rho \cos \varphi & 0 \\ 0 & 0 & 1\end{bmatrix}\]

For the determinant, we have:

\[\det J_{\tau}\left(\begin{bmatrix}\rho \\ \varphi \\ z \end{bmatrix}\right) = 1 \cdot (\rho \cos^2 \varphi + \rho \sin^2 \varphi) = \rho\]

Since \(\rho \ne 0\), we know that the columns of \(J_{\tau}\) form a local coordinate basis:

\[\boldsymbol{\rho} = \begin{bmatrix}\cos \varphi \\ \sin \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{\varphi} = \begin{bmatrix}-\rho \sin \varphi \\ \rho \cos \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{z} = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\]

We just need to normalize them:

\[\boldsymbol{\hat{\rho}} = \begin{bmatrix}\cos \varphi \\ \sin \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi \\ 0\end{bmatrix} \qquad \boldsymbol{\hat{z}} = \begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\]

Conventions#

Some people prefer to use \(\theta\) or \(\varphi\) for the azimuth.

Additionally, a single point has infinitely many possible values for its azimuth, since adding or subtracting a multiple of \(2\pi\) to an angle has no effect. However, in order to have a coordinate system, coordinates must be unique. This means that the possible values for \(\phi\) must be restricted. Common conventions for the range of \(\phi\) are \([0; 2\pi)\) and \((-\pi, \pi]\).