Cylindrical coordinates and their orthonormal frame

Away from the $z$ -axis, cylindrical coordinates are

x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=z, \qquad r=\sqrt{x^2+y^2}>0.

The angle in radians is taken modulo $2\pi$ , or on a local angular interval. The corresponding orthonormal frame is

e_r=(\cos\theta,\sin\theta,0),\quad e_\theta=(-\sin\theta,\cos\theta,0),\quad e_z=(0,0,1).

Its angular derivatives are $\partial_\theta e_r=e_\theta$ and $\partial_\theta e_\theta=-e_r$ .

Scalar derivatives and volume

The chain rule gives

\partial_x=\cos\theta\,\partial_r-\frac{\sin\theta}{r}\partial_\theta, \qquad \partial_y=\sin\theta\,\partial_r+\frac{\cos\theta}{r}\partial_\theta.

Consequently $\nabla f=e_r\partial_r f+e_\theta r^{-1}\partial_\theta f+e_z\partial_z f$ . The coordinate Jacobian has absolute determinant $r$ , so Euclidean volume is $r\,dr\,d\theta\,dz$ .

The axis

At $r=0$ , the angle and horizontal frame vectors are not defined. A field may nevertheless extend smoothly there; this is checked through its Cartesian components.

References

MIT 6.013, Differential operators in cylindrical and spherical coordinates.

Fluid components

In fluid mechanics these coordinates distinguish radial velocity, swirl, and axial velocity. Axisymmetry means that the cylindrical components are independent of the angle, even though their moving basis is not.