26.15 The metric tensor
second-order tensorT. Using the outer product notation in (26.23), we may write
Tin three different ways:
T=Tijei⊗ej=Tijei⊗ej=Tijei⊗ej,whereTij,Tij andTijare called thecontravariant, mixedandcovariantcom-
ponents ofTrespectively. It is important to remember that these three sets of
quantities form the components of thesametensorTbut refer to different (tensor)
bases made up from the basis vectors of the coordinate system. Again, if we are
using Cartesian coordinates then all three sets of components are identical.
We may generalise the above equation to higher-order tensors. Componentscarrying only superscripts or only subscripts are referred to as the contravariant
and covariant components respectively; all others are called mixed components.
26.15 The metric tensorAny particular curvilinear coordinate system is completely characterised at each
point in space by the nine quantities
gij=ei·ej, (26.56)which, as we will show, are the covariant components of a symmetric second-order
tensorgcalled themetric tensor.
Since an infinitesimal vector displacement can be written asdr=duiei, we findthat the square of the infinitesimal arc length (ds)^2 can be written in terms of the
metric tensor as
(ds)^2 =dr·dr=duiei·dujej=gijduiduj. (26.57)It may further be shown that the volume elementdVis given by
dV=√
gdu^1 du^2 du^3 , (26.58)wheregis the determinant of the matrix [gij], which has the covariant components
of the metric tensor as its elements.
If we compare equations (26.57) and (26.58) with the analogous ones in section10.10 then we see that in the special case where the coordinate system is orthogonal
(so thatei·ej=0fori=j) the metric tensor can be written in terms of the
coordinate-system scale factorshi,i=1, 2 ,3as
gij={
h^2 i i=j,0 i=j.Its determinant is then given byg=h^21 h^22 h^23.