Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

10.10 General curvilinear coordinates

As indicated earlier, the contents of this section are more formal and technically

complicated than hitherto. The section could be omitted until the reader has had

some experience of using its results.

Cylindrical and spherical polars are just two examples of what are called

general curvilinear coordinates. In the general case, the position of a pointP

having Cartesian coordinatesx, y, zmay be expressed in terms of the three

curvilinear coordinatesu 1 ,u 2 ,u 3 ,where

x=x(u 1 ,u 2 ,u 3 ),y=y(u 1 ,u 2 ,u 3 ),z=z(u 1 ,u 2 ,u 3 ),

and similarly

u 1 =u 1 (x, y, z),u 2 =u 2 (x, y, z),u 3 =u 3 (x, y, z).

We assume that all these functions are continuous, differentiable and have a

single-valued inverse, except perhaps at or on certain isolated points or lines,

so that there is a one-to-one correspondence between thex, y, zandu 1 ,u 2 ,u 3

systems. Theu 1 -,u 2 -andu 3 - coordinate curves of a general curvilinear system

are analogous to thex-,y-andz- axes of Cartesian coordinates. The surfaces

u 1 =c 1 ,u 2 =c 2 andu 3 =c 3 ,wherec 1 ,c 2 ,c 3 are constants, are called the

coordinate surfacesand each pair of these surfaces has its intersection in a curve

called acoordinate curveorline(see figure 10.11).

If at each point in space the three coordinate surfaces passing through the point

meet at right angles then the curvilinear coordinate system is calledorthogonal.

For example, in spherical polarsu 1 =r,u 2 =θ,u 3 =φand the three coordinate

surfaces passing through the point (R,Θ,Φ) are the spherer=R, the circular

coneθ= Θ and the planeφ= Φ, which intersect at right angles at that

point. Therefore spherical polars form an orthogonal coordinate system (as do

cylindrical polars).

Ifr(u 1 ,u 2 ,u 3 ) is the position vector of the pointPthene 1 =∂r/∂u 1 is a vector

tangent to theu 1 -curve atP(for whichu 2 andu 3 are constants) in the direction

of increasingu 1. Similarly,e 2 =∂r/∂u 2 ande 3 =∂r/∂u 3 are vectors tangent to

theu 2 -andu 3 - curves atPin the direction of increasingu 2 andu 3 respectively.

Denoting the lengths of these vectors byh 1 ,h 2 andh 3 ,theunitvectors in each of

these directions are given by

eˆ 1 =

1

h 1

∂r

∂u 1

, ˆe 2 =

1

h 2

∂r

∂u 2

, ˆe 3 =

1

h 3

∂r

∂u 3

,

whereh 1 =|∂r/∂u 1 |,h 2 =|∂r/∂u 2 |andh 3 =|∂r/∂u 3 |.

The quantitiesh 1 ,h 2 ,h 3 are the scale factors of the curvilinear coordinate

system. The element of distance associated with an infinitesimal changeduiin

one of the coordinates ishidui. In the previous section we found that the scale