Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.10 GENERAL CURVILINEAR COORDINATES

In the last step we have used the chain rule for partial differentiation. Thereforeei·j=1
ifi=j,andei·j= 0 otherwise. Hence{ei}and{j}are reciprocal systems of vectors.

We now derive expressions for the standard vector operators inorthogonal

curvilinear coordinates. Despite the useful properties of the non-unit bases dis-

cussed above, the remainder of our discussion in this section will be in terms of

the unit basis vectors{eˆi}. The expressions for the vector operators in cylindrical

and spherical polar coordinates given in tables 10.2 and 10.3 respectively can be

found from those derived below by inserting the appropriate scale factors.

Gradient

The changedΦ in a scalar field Φ resulting from changesdu 1 ,du 2 ,du 3 in the

coordinatesu 1 ,u 2 ,u 3 is given by, from (5.5),

dΦ=

∂Φ

∂u 1

du 1 +

∂Φ

∂u 2

du 2 +

∂Φ

∂u 3

du 3.

For orthogonal curvilinear coordinatesu 1 ,u 2 ,u 3 we find from (10.57), and com-

parison with (10.27), that we can write this as

dΦ=∇Φ·dr, (10.59)

where∇Φ is given by

∇Φ=

1

h 1

∂Φ

∂u 1

eˆ 1 +

1

h 2

∂Φ

∂u 2

ˆe 2 +

1

h 3

∂Φ

∂u 3

ˆe 3. (10.60)

This implies that the del operator can be written

∇=

ˆe 1

h 1

∂

∂u 1

+

ˆe 2

h 2

∂

∂u 2

+

ˆe 3

h 3

∂

∂u 3

.

Show that for orthogonal curvilinear coordinates∇ui=ˆei/hi. Hence show that the two

sets of vectors{eˆi}and{ˆi}are identical in this case.

Letting Φ =uiin (10.60) we find immediately that∇ui=ˆei/hi. Therefore|∇ui|=1/hi,and
soˆi=∇ui/|∇ui|=hi∇ui=ˆei.

Divergence

In order to derive the expression for the divergence of a vector field in orthogonal

curvilinear coordinates, we must first write the vector field in terms of the basis

vectors of the coordinate system:

a=a 1 ˆe 1 +a 2 ˆe 2 +a 3 ˆe 3.

The divergence is then given by

∇·a=

1

h 1 h 2 h 3

[

∂

∂u 1

(h 2 h 3 a 1 )+

∂

∂u 2

(h 3 h 1 a 2 )+

∂

∂u 3

(h 1 h 2 a 3 )

]

.

(10.61)