as before.

In the unit-vector basis, the gradient vector is

1 ‚φ 1 ‚φ 1 ‚φ

grad φ ≡ φ= e1 + e2 + e3 , (8.7)

h 1 ‚x1 h 2 ‚x2 h 3 ‚x3

so that

‚φ 1 ‚φ ‚φ

dx + 2 dx2 + 3 dx3 ,

(grad φ) · dr = (8.8)

‚x1 ‚x ‚x

which is the change in the value φ due the displacement.

8.1. CURVILINEAR CO-ORDINATES 219

The numbers (h1 dx1 , h2 dx2 , h3 dx3 ) are often called the physical compo-

nents of the displacement dr, to distinguish them from the numbers (dx1 , dx2 , dx3 )

which are the co-ordinate components of dr. The physical components of a

displacent vector all have the dimensions of length. The co-ordinate compo-

nents may have di¬erent dimensions and units for each component. In plane

polar co-ordinates, for example, the units will be meters and radians. This

distinction extends to the gradient itself: the co-ordinate components of an

electric ¬eld expressed in polar co-ordinates will have units of volts per me-

ter and volts per radian for the radial and angular components, respectively.

The factor 1/hθ = r ’1 serves to convert the latter to volts per meter.

The divergence

The divergence of a vector ¬eld A is de¬ned to be the ¬‚ux of A out of an

in¬nitesimal region, divided by volume of the region.

h3dx 3

h2dx 2

h1dx 1

Flux out of an in¬nitesimal volume with sides of length h1 dx1 , h2 dx2 , h3 dx3 .

In the ¬gure, the ¬‚ux out of the two end faces is

‚(A1 h2 h3 )

dx2 dx3 A1 h2 h3 |(x1 +dx1 ,x2 ,x3 ) ’ A1 h2 h3 |(x1 ,x2 ,x3 ) ≈ dx1 dx2 dx3

.

‚x1

(8.9)

Adding the contributions from the other two pairs of faces, and dividing by

the volume, h2 h2 h3 dx1 dx2 dx3 , gives

1 ‚ ‚ ‚

div A = (h2 h3 A1 ) + (h1 h3 A2 ) + (h1 h2 A3 ) . (8.10)

h1 h2 h3 ‚x1 ‚x2 ‚x3

220 CHAPTER 8. SPECIAL FUNCTIONS I

·A, although

Note that in curvilinear coordinates div A is no longer simply

one often writes it as such.

The curl

The curl of a vector ¬eld A is a vector whose component in the direction of

the normal to an in¬nitesimal area element, is line integral of A round the

in¬nitsimal area, divided by the area.

e3

h2dx 2

h dx 1

1

Line integral round in¬nitesimal area with sides of length h1 dx1 , h2 dx2 , and

normal e3 .

The third component is, for example,

1 ‚h2 A2 ‚h1 A1

’

(curl A)3 = . (8.11)

‚x1 ‚x2

h1 h2

The other two components are found by cyclically permuting 1 ’ 2 ’ 3 ’ 1

in this formula. The curl is thus is no longer equal to — A, although it is

common to write it as if it were.

Note that the factors of hi are disposed so that the vector identies

curl grad • = 0, (8.12)

and

div curl A = 0, (8.13)

continue to hold for any scalar ¬eld •, and any vector ¬eld A.

8.2. SPHERICAL HARMONICS 221

8.1.2 The Laplacian in Curvilinear Co-ordinates

The Laplacian acting on scalars, is “div grad”, and is therefore

1 ‚ h2 h3 ‚φ ‚ h1 h3 ‚φ ‚ h1 h2 ‚φ

2

φ= + + .

h1 h2 h3 ‚x1 h1 ‚x1 ‚x2 h2 ‚x2 ‚x3 h3 ‚x3

(8.14)

This formula is worth commiting to memory.

When the Laplacian is to act on vectors, we must use

2

A = grad div A ’ curl curl A. (8.15)

In curvilinear co-ordinates this is no longer equivalent to the Laplacian acting

on each component of A, treating it as if it were a scalar.

In spherical polars the Laplace operator acting on the scalar ¬eld φ is

‚2•

1‚ 2 ‚• 1 ‚ ‚• 1

2

•= 2 r +2 sin θ +2 2

r sin θ ‚φ2

r ‚r ‚r r sin θ ‚θ ‚θ

1 ‚ 2 (r•) 1 ‚2•

1 1‚ ‚•

= +2 sin θ +

sin2 θ ‚φ2

r ‚r2 r sin θ ‚θ ‚θ

ˆ

1 ‚ 2 (r•) L2

’ 2 •,

= (8.16)

r ‚r2 r

where

2

ˆ 2 = ’ 1 ‚ sin θ ‚ ’ 1 ‚ ,

L (8.17)

‚θ sin2 θ ‚φ2

sin θ ‚θ

is (after multiplication by h2 ) the operator representing the square of the

¯

angular momentum in quantum mechanics.

In cylindrical polars the Laplacian is

1 ‚2 ‚2

1‚ ‚

2

= r+ + . (8.18)

r ‚r ‚r r 2 ‚θ2 ‚z 2