Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.10 GENERAL CURVILINEAR COORDINATES

z

x

i y

j

k

O

P

u 2 =c 2

u 1 =c 1

u 3 =c 3

u 1

u 2

u 3

ˆ 1

ˆ 2

ˆ 3

ˆe 1

eˆ 2

ˆe 3

Figure 10.11 General curvilinear coordinates.

factors for cylindrical and spherical polar coordinates were

for cylindrical polars hρ=1, hφ=ρ, hz=1,

for spherical polars hr=1, hθ=r, hφ=rsinθ.

Although the vectorse 1 ,e 2 ,e 3 form a perfectly good basis for the curvilinear

coordinate system, it is usual to work with the corresponding unit vectorseˆ 1 ,eˆ 2 ,

ˆe 3. For an orthogonal curvilinear coordinate system these unit vectors form an

orthonormal basis.

An infinitesimal vector displacement in general curvilinear coordinates is given

by, from (10.19),

dr=

∂r

∂u 1

du 1 +

∂r

∂u 2

du 2 +

∂r

∂u 3

du 3 (10.55)

=du 1 e 1 +du 2 e 2 +du 3 e 3 (10.56)

=h 1 du 1 eˆ 1 +h 2 du 2 eˆ 2 +h 3 du 3 ˆe 3. (10.57)

Inthecaseoforthogonalcurvilinear coordinates, where theˆeiare mutually

perpendicular, the element of arc length is given by

(ds)^2 =dr·dr=h^21 (du 1 )^2 +h^22 (du 2 )^2 +h^23 (du 3 )^2. (10.58)

The volume element for the coordinate system is the volume of the infinitesimal

parallelepiped defined by the vectors (∂r/∂ui)dui=duiei=hiduiˆei,fori=1, 2 ,3.