We have to use the cylindrical coordinates when the problem is cylindrical symmetry. For example, the eigenmode in a circular waveguide. Althrough there is a general theory for the differential operators in general curvilinear coordinates, it it very difficult to be understanded. There is also the formular for the differential operators in cylindrical coordinates. However it is too strange to be memoried. The differential operators in cylindrical coordinates is obtained from the general rule in this article.
A point M is defined by the cylindrical coordinates \(r,\theta,z\) in a local orthonormal basis \(( e_r,e_\theta, e_z)\). The vector OM is:
\(\vec{OM}=r \cos\theta e_x+r \sin\theta e_y+z e_z\)
Derivation of OM with the cylindrical coordinates gives:
\(\partial_r\vec{OM}=\cos\theta\hat e_x+\sin\theta\hat e_y=e_r\)
\(\partial_\theta\vec{OM}=r(-\cos\theta e_x+\cos\theta e_y=r e_\theta\)
\(\partial_z\vec{OM}=\vec e_z\)
It is clear that
\(\partial_\theta e_r=e_\theta\, \, \partial_\theta e_\theta=-e_r\)
Define the scale factors as
\(h_1=|\partial_r\vec{OM}|=|e_r|=1,\,\,\, h_2=|\partial_\theta\vec{OM}|=|r e_\theta|=r,\,\,\, h_3=|\partial_z|=1\)
and their product \(J=h_1h_2h_3=r\)
So, the scale factor is 1 when the motion is along `\vec(OM)` while the scale factor is `|\vec (OM)|` when the motion is perpendicular to OM.
Sure, here’s a table summarizing the scale factors in Cartesian, cylindrical, and spherical coordinates:
Certainly! Here’s the updated table with the header changed as requested:
| Coordinate System | `h_1` | `h_2` | `h_3` |
|---|---|---|---|
| Cartesian | 1 | 1 | 1 |
| Cylindrical | 1 | r | 1 |
| Spherical | 1 | r | `r sin theta` |
Gradient
The gradient of a scalar function F is the vector whose components are the partial derivatives of F with respect to each variable. In arbitrary orthogonal coordinates systems, the gradient is expressed in terms of the scale factors as follows:
\((\nabla F)_i = \frac{1}{h_i}\partial_iF\)
Applied it in the cylindrical coordinates,
\(\nabla F=\frac{1}{h_1}\partial_r F e_r+\frac{1}{h_2}\partial_\theta F e_\theta+\frac{1}{h_3}\partial_z F e_z=\partial_r F e_r+\frac{1}{r}\partial_\theta F e_\theta+\partial_z F e_z\)
Divergence
The divergence of a vector function \(F_i\) produces a scalar value \(\nabla \cdot F\) representing the volume density of the outward flux of the vector field form an infinitesimal volume around a given point. In terms of scale factors, it is expressed as follows:
\(\nabla \cdot F = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i}F_i\Biggl)\)
Applied it in cylindrical coordinates,
\(\nabla \cdot F = \frac{1}{r}(\partial_r(\frac{r}{1}F_r)+\partial_\theta(\frac{r}{r}F_\theta)+\partial_z(\frac{r}{1}F_z))=\frac{1}{r}\partial_r r F_r+\frac{1}{r}\partial_\theta F_\theta+\partial_z F_z\)
Curl
The curl of a vector function \(F_i\) at a point is represented by a vector whose length and direction denote the magnitude and the axis of the maximum circulation. In 3 dememsion, the curl is written in terms of the scale factors
\(\nabla \times F=\frac{1}{J}\begin{vmatrix}
h_1e_r & h_2e_\theta & h_3e_z\\
\partial_r & \partial_\theta & \partial_z \\
h_1 F_r & h_2 F_\theta & h_3 F_z\end{vmatrix}=
\frac{1}{r}\begin{vmatrix}
e_r & r e_\theta & e_z\\
\partial_r & \partial_\theta & \partial_z \\
F_r & r F_\theta & F_z\end{vmatrix}\)
The detail formula is here.