Cylindrical coordinates

We extend from polar coordinates from ${\mathbb{R}}^2$ to ${\mathbb{R}}^3$ with cylindrical coordinates. We keep our radial distance $r$, principal angle $\theta$ on the $xy$ plane. We add a new term $z$ (same as in Cartesian coordinates). Our fundamental terms are:

$$x=r\cos \theta$$ $$y=r\sin \theta$$ $$z=z$$ $$\tan \theta =\frac{y}{x}$$

The differential element is: $rdzdrd\theta$.

Triple integrals

An important application of cylindrical coordinates is in computing triple integrals where the region of integration is complicated.

Addendums

To transform from cylindrical to rectangular coordinates, we have a transformation matrix:

$$\text{M}=\left[\begin{array}{ccccc}\cos \theta & & -\sin \theta & & 0\\ \sin \theta & & \cos \theta & & 0\\ 0& & 0& & 1\end{array}\right]$$

${\text{M}}^T$ gives us the opposite conversion.

See also

• Spherical coordinates