Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Thus, by inverting the matrixGin (26.60), we find that the elementsgijare given in
cylindrical polar coordinates by

Gˆ=[gij]=





100

01 /ρ^20

001



.

So far we have not considered the components of the metric tensorgjiwith one

subscript and one superscript. By analogy with (26.56), these mixed components

are given by

gij=ei·ej=δij,

and so the components ofgjiare identical to those ofδji.Wemaytherefore

consider theδijto be the mixed components of the metric tensorg.

26.16 General coordinate transformations and tensors

We now discuss the concept of general transformations from one coordinate

system,u^1 ,u^2 ,u^3 , to another,u′^1 ,u′^2 ,u′^3. We can describe the coordinate transform

using the three equations

u′

i

=u′

i

(u^1 ,u^2 ,u^3 ),

fori=1, 2 ,3, in which the new coordinatesu′ican be arbitrary functions of the old

onesuirather than just represent linear orthogonal transformations (rotations)

of the coordinate axes. We shall assume also that the transformation can be

inverted, so that we can write the old coordinates in terms of the new ones as

ui=ui(u′^1 ,u′^2 ,u′^3 ),

As an example, we may consider the transformation from spherical polar to

Cartesian coordinates, given by

x=rsinθcosφ,

y=rsinθsinφ,

z=rcosθ,

which is clearly not a linear transformation.

The two sets of basis vectors in the new coordinate system,u′^1 ,u′^2 ,u′^3 , are given

as in (26.55) by

e′i=

∂r

∂u′i

and e′

i

=∇u′

i

. (26.66)

Considering the first set, we have from the chain rule that

∂r

∂uj

=

∂u′i

∂uj

∂r

∂u′i

,