Cylindrical coordinates

Cylindrical coordinates specify a point in 3D space by $(\rho ,\varphi ,z)$:

Image: Cylindrical coordinate system (ISO 31-11 convention), public domain

• $\rho \ge 0$: perpendicular distance from the $z$-axis,
• $\varphi$: angular position around the $z$-axis, measured from the $+x$-axis ($0\le \varphi <2\pi$),
• $z$: height (same as Cartesian).

Cylindrical coordinates are 2D polar coordinates in the $xy$-plane, extruded along $z$. They’re the natural choice when a problem has rotational symmetry about an axis.

Conversion to Cartesian

$x=\rho \cos \varphi ,y=\rho \sin \varphi ,z=z.$

Going the other way: $\rho =\sqrt{x^2+y^2}$, $\varphi =atan2(y,x)$.

Unit vectors

$\hat{\mathbf{\rho }},\hat{\mathbf{\varphi }},\hat{\mathbf{z}},$

with

• $\hat{\mathbf{\rho }}$ pointing radially outward (in the $xy$-plane), away from the $z$-axis,
• $\hat{\mathbf{\varphi }}$ tangent to a horizontal circle around the $z$-axis, counterclockwise viewed from above,
• $\hat{\mathbf{z}}$ same as $\hat{\mathbf{k}}$.

Cartesian relations:

$\hat{\mathbf{\rho }}=\cos \varphi \hat{\mathbf{i}}+\sin \varphi \hat{\mathbf{j}},\hat{\mathbf{\varphi }}=-\sin \varphi \hat{\mathbf{i}}+\cos \varphi \hat{\mathbf{j}},\hat{\mathbf{z}}=\hat{\mathbf{k}}.$

Crucially, $\hat{\mathbf{\rho }}$ and $\hat{\mathbf{\varphi }}$ are position-dependent — they rotate as $\varphi$ changes. Unlike Cartesian $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$, they cannot be pulled outside a derivative or integral without care.

Differential elements

Differential length:

$d\mathbf{l}=d\rho \hat{\mathbf{\rho }}+\rho d\varphi \hat{\mathbf{\varphi }}+dz\hat{\mathbf{z}}.$

The factor of $\rho$ multiplying $d\varphi$ is essential. A small angular change $d\varphi$ at radius $\rho$ corresponds to an arc-length displacement of $\rho d\varphi$, not just $d\varphi$. Forgetting this $\rho$ factor is the most common error in cylindrical-coordinate integration.

Differential volume:

$dv=\rho d\rho d\varphi dz.$

The $\rho$ factor is the Jacobian of the change of variables from Cartesian to cylindrical.

Surface element on a cylinder of fixed radius ${\rho }_0$:

$d\mathbf{s}={\rho }_{0}d\varphi dz\hat{\mathbf{\rho }}.$

Outward normal points radially.

When cylindrical is right

• Geometry with rotational symmetry around an axis: a wire, a coil, a circular waveguide, a coaxial cable.
• Fields produced by infinite line charges or currents.
• Fluid flow in a pipe.

The classical example: a uniformly charged infinite wire produces an electric field $\mathbf{E}=(\lambda /(2\pi {ϵ}_0\rho ))\hat{\mathbf{\rho }}$. Pure radial dependence, no $\varphi$ or $z$ — three integrals collapse to one.

Distance between two points

For two points $P_1=({\rho }_1,{\varphi }_1,z_1)$ and $P_2=({\rho }_2,{\varphi }_2,z_2)$, the Euclidean distance is

$d=\sqrt{{\rho }_1^2+{\rho }_2^2-2{\rho }_1{\rho }_2\cos ({\varphi }_2-{\varphi }_1)+(z_2-z_1{)}^2}.$

The cross term $-2{\rho }_1{\rho }_2\cos ({\varphi }_2-{\varphi }_1)$ comes from the law of cosines applied in the $xy$-plane: the two radial vectors meet at angle ${\varphi }_2-{\varphi }_1$. When ${\varphi }_1={\varphi }_2$ (same azimuth), this collapses to $|{\rho }_2-{\rho }_1|$ plus the $z$ piece. When ${\rho }_1={\rho }_2=\rho$ and $z_1=z_2$ (two points on a horizontal circle), it becomes $2\rho \sin (({\varphi }_2-{\varphi }_1)/2)$, the chord length.

This formula is what you use when computing the distance vector magnitude between a source and field point in Coulomb’s law integrals that are most naturally set up in cylindrical coordinates.

Bridge with Cartesian and spherical

• Setting $z=0$ reduces to 2D polar coordinates in the plane.
• Spherical $(r,\theta ,\varphi )$: same $\varphi$ as cylindrical (azimuthal angle); $r$ and $\theta$ replace cylindrical $\rho$ and $z$ via $\rho =r\sin \theta$, $z=r\cos \theta$.