Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The symmetry of a problem often makes one coordinate system far more convenient than another: a line charge calls for cylindrical coordinates, a point charge for spherical ones. All the coordinate systems we use are orthogonal: at every point the three coordinate directions are mutually perpendicular. For a coordinate system with parameters $(u_{1}, u_{2}, u_{3})$ we need to know three things for each coordinate (Table 2.3.1).

Table 2.3.1. What has to be known about an orthogonal coordinate system.

Parameter definitions	Unit vectors	Scale factors (metric coefficients)
$u_{1}$	$a_{1}$	$h_{1}$
$u_{2}$	$a_{2}$	$h_{2}$
$u_{3}$	$a_{3}$	$h_{3}$

The parameter definitions express the Cartesian coordinates as functions of $(u_{1}, u_{2}, u_{3})$ , so that the position vector $r = x a_{x} + y a_{y} + z a_{z}$ becomes a function $r (u_{1}, u_{2}, u_{3})$ .
The unit vector $a_{i}$ points in the direction in which $P$ moves when $u_{i}$ alone is increased. The three are mutually perpendicular and ordered so that $a_{1} \times a_{2} = a_{3}$ (right-handed).
The scale factor $h_{i}$ converts a change of the coordinate, $d u_{i}$ , into a distance, $h_{i} d u_{i}$ . This is needed because a coordinate need not be a length: an angle is not (Figure 2.3.3).

Coordinate curves of an orthogonal coordinate system in a plane (here u_1 is a distance from a centre and u_2 an angle, as in polar coordinates). Increasing u_1 by du_1 moves \mathrm P along the green line by h_1\,du_1, and increasing u_2 by du_2 moves it along the blue curve by h_2\,du_2. The shaded cell has sides h_1du_1 and h_2du_2, and the unit vectors \vec a_1, \vec a_2 are tangent to the coordinate curves. At another point they point in different directions. — **Figure 2.3.3.** Coordinate curves of an orthogonal coordinate system in a plane (here $u_{1}$ is a distance from a centre and $u_{2}$ an angle, as in polar coordinates). Increasing $u_{1}$ by $d u_{1}$ moves $P$ along the green line by $h_{1} d u_{1}$ , and increasing $u_{2}$ by $d u_{2}$ moves it along the blue curve by $h_{2} d u_{2}$ . The shaded cell has sides $h_{1} d u_{1}$ and $h_{2} d u_{2}$ , and the unit vectors $a_{1}$ , $a_{2}$ are tangent to the coordinate curves. At another point they point in different directions.

All three are contained in one relation. When only $u_{i}$ changes, by $d u_{i}$ , the point moves by $d r = (\partial r / \partial u_{i}) d u_{i}$ . The direction of this vector is $a_{i}$ and its length is $h_{i} d u_{i}$ . Therefore the first-order partial derivative of the position vector with respect to a parameter equals the scale factor times the unit vector:

Scale factors and unit vectors

\frac{\partial r}{\partial u _{1}} \frac{\partial r}{\partial u _{3}} = h_{1} a_{1}, \frac{\partial r}{\partial u _{2}} = h_{2} a_{2}, = h_{3} a_{3}, h_{i} = \frac{\partial r}{\partial u _{i}} . (2.3.5)

A general small step changes all three coordinates. By the chain rule $d r = \sum_{i} (\partial r / \partial u_{i}) d u_{i}$ , which gives the differential length (differential displacement) vector

d ℓ = h_{1} d u_{1} a_{1} + h_{2} d u_{2} a_{2} + h_{3} d u_{3} a_{3} . (2.3.6)

Increasing each coordinate by $d u_{i}$ sweeps out a small box with perpendicular edges $h_{1} d u_{1}$ , $h_{2} d u_{2}$ , $h_{3} d u_{3}$ . Its faces are the differential areas, and as oriented areas (normal vector times area) they are

d S_{1} d S_{2} d S_{3} = a_{1} h_{2} h_{3} d u_{2} d u_{3}, = a_{2} h_{3} h_{1} d u_{3} d u_{1}, = a_{3} h_{1} h_{2} d u_{1} d u_{2}, (2.3.7)

where $d S_{i}$ is the face on which $u_{i}$ is constant. Its volume is the differential volume

d V = h_{1} d u_{1} h_{2} d u_{2} h_{3} d u_{3} = h_{1} h_{2} h_{3} d u_{1} d u_{2} d u_{3} . (2.3.8)

Equations (2.3.6) to (2.3.8) are what we need to write line, surface and volume integrals in any orthogonal coordinate system. In Section 2.3.10 the same three scale factors will also give the gradient, divergence, curl and Laplacian in curvilinear coordinates.

In plain words

A scale factor answers: “if I change this coordinate by a tiny amount, how many metres do I move?” For a coordinate that is already a length the answer is $h = 1$ . For an angle the answer depends on how far from the centre we are. Once the three $h_{i}$ are known, every length, area and volume element is a product of edges $h_{i} d u_{i}$ .

2.3.2 Coordinate systems