Appendix: Orthogonal Curvilinear Coordinates

Appendix: Orthogonal Curvilinear Coordinates

Notes:

Most of the material presented in this chapter is taken from Anupam, G. (Classical
Electromagnetism in a Nutshell 2012, (Princeton: New Jersey)), Chap. 2, and Weinberg,
S. (Gravitation and Cosmology: Principles and Applications of the General Theory of
Relativity 1972, (Wiley: New York), Chap. 8.

We define the infinitesimal spatial displacement vector dx in a given orthogonal
coordinate system with

dx = dxiei, (II.1)

where the Einstein summation convention was used, dxi is a contravariant component
and ei is a basis vector ( i = 1,2, 3 ). The length interval ds is thus given by

ds2 = dx ⋅ dx (II.2)

( ) ( )= dxiei ⋅ dx je j
( )= ei ⋅ e j dxidx j

= gijdxidx j

= dxidxi,

where the orthogonality of the coordinate system is specified by ei ⋅ e j = gij with the

metric tensor gij = 0 when i ≠ j , and dxi is the covariant component. Please note that

basis vectors are not unit vectors, i.e., ei ⋅ ei ≠ 1 in general. Equation (II.2) can be used to
similarly define the inner product between any two vectors with

A ⋅ B = gij Ai B j (II.3)
= Ai Bi .

Since the covariant and contravariant components are generally different from one
another in non-Cartesian coordinate systems, it is often more desirable to introduce a new

set of so-called ordinary or physical components that preserve the inner product without
explicitly bringing in both types of components or the metric tensor.

We start by rewriting equation (II.2) as

( ) ( ) ( )ds2 = h1du1 2 + h2du2 2 + h3du3 2 (II.4)

I

( )for the orthogonal coordinate system u1,u2,u3 . A comparison with equation (II.2)

reveals that hi2 = gii . For example, Cartesian coordinates have h1 = h2 = h3 = 1 ,

cylindrical coordinates (ρ,θ, z) have h1 = h3 = 1, h2 = ρ , and spherical coordinates
(r,θ,φ) have h1 = 1, h2 = r , and h3 = r sin(θ ) . Going back to equation (II.3) for the

inner product, we now define the physical coordinates Ai of a vector A such that

A ⋅ B ≡ Ai Bi, (II.5)

where the use of subscripts has no particular meaning (i.e., a subscript does not imply a
covariant component). A comparison with equation (II.3) implies that the physical

components are related to the covariant and contravariant components through

Ai = hi Ai = hi−1Ai . (II.6)

The first thing we should notice is that the physical components allow the use of a unit
basis eˆi since

ei ⋅ e j = gij (II.7)

= hihjδ ij

( )= hihj eˆi ⋅ eˆ j
( )= (hieˆi )⋅ hjeˆ j .

In fact, we could have alternatively justified the introduction of the physical components
by the desire to use a unit basis with

dx = h1du1eˆ1 + h2du2eˆ 2 + h3du3eˆ 3 (II.8)
= dx1eˆ1 + dx2eˆ 2 + dx3eˆ 3

or in general

A = A1eˆ1 + A2eˆ 2 + A3eˆ 3. (II.9)

It should now be clear that what we usually specify as coordinates (e.g., (ρ,θ, z) and
(r,θ,φ) ) correspond to the contravariant components of dx , while the physical

coordinates are those for which the components of dx have units of length (e.g.,

(dρ, ρdθ,dz) and (dr,rdθ,r sin(θ )dφ) ).

We now define the different differential operators using the physical coordinates, starting
with the gradient. To do so, we first consider the differential of a scalar function f

II

df = ∂f dui
∂ui

≡ ∇f ⋅ dx

∑= ∇f ⋅ ⎛ hi du i eˆ i ⎞ (II.10)
⎜⎝ ⎠⎟
i

∑= hidui∇f ⋅ eˆi,
i

which from the first and last equations implies that

∇f = 1 ∂f eˆ1 + 1 ∂f eˆ 2 + 1 ∂f eˆ 3. (II.11)
h1 ∂u1 h2 ∂u 2 h3 ∂u 3

This leads to the following relations for the cylindrical and spherical coordinate systems

∇f = ∂f eˆ ρ + 1 ∂f eˆθ + ∂f eˆ z
∂ρ ρ ∂θ ∂z
(II.12)
∂f 1 ∂f 1 ∂f
∇f = ∂r eˆ r + r ∂θ eˆθ + ∂φ eˆφ ,
r sin(θ )

Review of mirneasFtphiegceutmirveea1ly,ti.acnFadolurcstoehnethcdeeivdpeivrtgesregnecnecoefthaevoercetmor we consider an infinitesimal cube, as shown

Top e

h2 du2 u2 = constant

e1 B h3 du3
h1 du1 h1 du1

u1 = constant h2 du2
Bottom

(a) Figure 1 - Infin(ibte)simal volume of
integration, where we do not

near coordinates. (a) Two-dimensional illustdraiftfieorne.n(tbia)teInbteetgwreaetinon regainodns fo.r derivation of
rgence and curl.

oordinates (u1, u2, u3) = (r, u, z), related to CartIeIsIian coordinates by x =
n u, and z = z,

∫V ∇ ⋅ Ad 3x = ∇ ⋅ Ah1h2h3du1du2du3 (II.13)
= ∫S A ⋅ n da,

which for a small enough cube we can write as

∫S A ⋅ n da = ⎣⎡ A1h2h3 ⎤⎦left du 2 du 3 + ⎣⎡ A2h1h3 ⎤⎦front du1du 3
right back

+ ⎡⎣ A3h1h2 ⎤⎦ top du1du 2 (II.14)
bottom

( ) ( ) ( )=⎡∂ ∂ ∂ ⎤ du1du 2du 3,
⎢⎣ ∂u1 A1h2h3 + ∂u 2 A2h1h3 + ∂u 3 A3h1h2 ⎦⎥

since in general hi can vary across the dimensions of the cube. A comparison with
equation (II.13) reveals that

1 ⎡∂ ∂ ∂ ⎤⎦⎥.
h1h2h3 ⎣⎢ ∂u1 ∂u 2 ∂u 3
( ) ( ) ( )∇ (II.15)
⋅A = A1h2h3 + A2h1h3 + A3h1h2

We then respectively have for cylindrical and spherical coordinates

( )∇ 1 ∂ + 1 ∂Aθ + ∂Az
⋅A = ρ ∂ρ ρ Ar ρ ∂θ ∂z

1 ∂ 1 ∂ 1 ∂ Aφ (II.16)
r2 ∂r ∂θ ∂φ
( )∇ ⋅A sin (θ ( (θ ) ) sin (θ
= r2 Ar + r ) sin Aθ + r ) .

The Laplacian is readily evaluated by setting A = ∇f and inserting equations (II.12) in
equations (II.16). We then have the corresponding relations

∇2 f = 1 ∂ ⎛ ρ ∂f ⎞ + 1 ∂2 f + ∂2 f
ρ ∂ρ ⎜⎝ ∂ρ ⎠⎟ ρ2 ∂θ 2 ∂z2
(II.17)
1 ∂ ∂f 1 ∂ ∂f 1 ∂2 f
∇2 f = r2 ∂r ⎝⎛⎜ r2 ∂r ⎠⎟⎞ + ∂θ ⎜⎝⎛ sin (θ ) ∂θ ⎠⎞⎟ + ∂φ 2
r2 sin(θ ) r2 sin2 (θ )

for cylindrical and spherical coordinates, respectively.

Finally, for the curl we use Stokes’ Theorem using an infinitesimal surface as shown in
Figure 2

∫ ( )(∇ × A)⋅nda
S
= (∇ × A)⋅ α h2h3du2du3eˆ1 + βh1h3du1du3eˆ 2 +γ h1h2du1du2eˆ 3 (II.18)

= !∫C A ⋅ dl,

IV

52 | Chapter 2 Review of mathematical concepts e

Top

e2 h2 du2 u2 = constant
e1
h3 du3
A h1 du1 B h1 du1

u1 = constant h2 du2
Bottom

(a) (b)

Figure 2 – Infinitesimal loop of integration for
Figure 2t.h2.e dCeurrivvialitnioenarocfotohredicnuartle,sp. r(oa)jeTcwtioo-ndiimn ethnesional illustration. (b) Integration regions for der
expressions for divergen.ce and curl.

where n = αeˆ1 + βeˆ2 + γ eˆ 3 . For this infinitesimal loop we can consider the different

( )projections Fonorthceylitnhrdereicauli,cuojo-rpdlianneastesan(du1w, urit2e, u(u3s)i=ng (rth,eu,fizr)s,t rtewloatetedrmtos Cofarttheesian coordinates

correspondinrgcToasyulo, ryex=parnsiionnus), and z = z,

2 d rA2 lr=2 dαu2⎣⎡ z .⎦⎤b2ottom
C top
!∫ { }ds= +⋅ d A+2hd2 du 2 front du 3
+ ⎣⎡ A3h3 ⎦⎤ back

{ }+β
Hence, ⎡⎣ A1h1 ⎦⎤top du1 + ⎡⎣ A3h3 ⎤⎦right du 3
bottom left

{ }hr = 1, ⎤⎦left
hu =+γr, ⎡⎣ A1hh1z⎦⎤=fbraocnkt1d.u1 + ⎡⎣ A2h2 right du 2

All vector field= αde⎡⎣⎢r−iv∂a∂ut3i(vAe2shc2 )a+n ∂ w( Ar3iht3t)e⎥⎤⎦n immediately (II.19) factors are

∂bue2 once the scale

We first quote a∂∂l++ulfγβ1the⎢⎣⎡⎢⎣⎡−e1∂∂u+∂r3∂ue(·s2Au·(1A·lht11,sh)1,−)a+∂n∂u∂d2∂u(d1A(eA3rh2i3hv)2e⎤⎦⎥)⎤⎦⎥th. em later:
∇f
= 1
h1

Equating equatio∇ns·(AII.1=8)ha1nhd12(hII3.19)∂w∂ue1 m(hu2sht 3haAv1e) + · · · ,

∂∂ (Ah23hA2 3⎤⎥⎦)eˆ−1 +∂h∂u11h33
( ) ( ) ( ) ( )∇ × A =∇h2×1h3A⎢⎡⎣ ∂=∂u2 ∂∂uu 32
A13h3 − (⎢⎣⎡h∂2∂uA3 2A) 1h1e1−+∂∂u· 1· ·A,3h3 ⎤ eˆ 2
h2h3 ⎥⎦
(II.20)
respec+tihv∇11eh2l2yf⎡⎣⎢w∂=∂rui1tehAf1o2hhr122thh−e3 ∂ ⎤
( ) ( )We ∂u 2 ∂A1h1 2eˆh3.3 s∂p∂uhfe1rica+l c·o·o·rdin. ate
h⎦⎥

then cyl∂inud1rical ahn1d systems

Here, Ai = A · ei , and the ellipses indicate terms that can be written down b

permutation of the indices V1, 2, and 3. These formulas apply to any orthogonal coo

system. Specific formulas for spherical polar and cylindrical coordinates can be o

by using eqs. (11.3) and (11.5), respectively.

⎡1 ∂ Az ∂ Aθ ⎤ ⎡ ∂ Aρ ∂ Az ⎤ 1 ⎡∂ ∂ Aρ ⎤
( )∇ × A ⎢⎣ ρ ∂θ ∂z ⎦⎥ eˆ ⎢ ∂z ∂ρ ⎥ eˆθ ρ ∂θ ⎥ eˆ z
= − ρ + ⎣ − + ⎢ ∂ρ ρ Aθ − ⎦
⎣
⎦

1 ⎡∂ ∂ Aθ ⎤ ⎡1 ∂ Ar 1 ∂ ⎤
( ) ( )∇ ⎢⎣ ∂θ ∂φ ⎥⎦ ∂φ r ∂r ⎥
× A = r sin (θ ) sin(θ ) Aφ − eˆ r + ⎢ r sin (θ ) − rAφ eˆθ (II.21)
⎣
⎦

( )+1 ⎡ ∂ rAθ − ∂ Ar ⎤ eˆφ .
r ⎢⎣ ∂r ∂θ ⎥⎦

VI