
Chapter 2 9
Curvilinear Coordinates, Vectors,
Operators, and Differential
Relations
29.1 Arbitrary Curvilinear Coordinate Systems
29.1.1 G eneral Nonorthogonal Curvilinear Coordinates
◮ Metric tensor. Arc length and volume elements in curvilinear coordinates.
The curvilinear coordinates x
1
, x
2
, x
3
are defined as functions of the rectangular Cartesian
coordinates x, y, z:
x
1
= x
1
(x, y, z), x
2
= x
2
(x, y, z), x
3
= x
3
(x, y, z).
Using these formulas, one can express x, y, z in terms of the curvilinear coordinates
x
1
, x
2
, x
3
as follows:
x = x(x
1
, x
2
, x
3
), y = y(x
1
, x
2
, x
3
), z = z(x
1
, x
2
, x
3
).
The metric tensor components g
ij
are determined by the formulas
g
ij
(x
1
, x
2