Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Calculate the elementsgijof the metric tensor for cylindrical polar coordinates. Hence

find the square of the infinitesimal arc length(ds)^2 and the volumedVfor this coordinate

system.

As discussed in section 10.9, in cylindrical polar coordinates (u^1 ,u^2 ,u^3 )=(ρ, φ, z)andso
the position vectorrof any pointPmay be written

r=ρcosφi+ρsinφj+zk.

From this we obtain the (covariant) basis vectors:

e 1 =

∂r

∂ρ

=cosφi+sinφj;

e 2 =

∂r

∂φ

=−ρsinφi+ρcosφj;

e 3 =

∂r

∂z

=k. (26.59)

Thus the components of the metric tensor [gij]=[ei·ej] are found to be

G=[gij]=





100

0 ρ^20

001



, (26.60)

from which we see that, as expected for an orthogonal coordinate system, the metric tensor
is diagonal, the diagonal elements being equal to the squares of the scale factors of the
coordinate system.
From (26.57), the square of the infinitesimal arc length in this coordinate system is given
by

(ds)^2 =gijduiduj=(dρ)^2 +ρ^2 (dφ)^2 +(dz)^2 ,

and, using (26.58), the volume element is found to be

dV=

√

gdu^1 du^2 du^3 =ρdρdφdz.

These expressions are identical to those derived in section 10.9.

We may also express the scalar product of two vectors in terms of the metric

tensor:

a·b=aiei·bjej=gijaibj, (26.61)

where we have used the contravariant components of the two vectors. Similarly,

using the covariant components, we can write the same scalar product as

a·b=aiei·bjej=gijaibj, (26.62)

where we have defined the nine quantitiesgij=ei·ej. As we shall show, they form

the contravariant components of the metric tensorgand are, in general, different

from the quantitiesgij. Finally, we could express the scalar product in terms of

the contravariant components of one vector and the covariant components of the

other,

a·b=aiei·bjej=aibjδji=aibi, (26.63)