26.15 THE METRIC TENSOR
where we have used the reciprocity relation (26.54). Similarly, we could write
a·b=aiei·bjej=aibjδji=aibi. (26.64)By comparing the four alternative expressions (26.61)–(26.64) for the scalarproduct of two vectors we can deduce one of the most useful properties of
the quantitiesgijandgij.Sincegijaibj=aibiholds for any arbitrary vector
componentsai, it follows that
gijbj=bi,which illustrates the fact that the covariant componentsgijofthemetrictensor
can be used tolower an index. In other words, it provides a means of obtaining
the covariant components of a vector from its contravariant components. By a
similar argument, we have
gijbj=bi,so that the contravariant componentsgijcan be used to perform the reverse
operation ofraising an index.
It is straightforward to show that the contravariant and covariant basis vectors,eiandeirespectively, are related in the same way as other vectors, i.e. by
ei=gijej and ei=gijej.We also note that, sinceeiandeiare reciprocal systems of vectors in three-
dimensional space (see chapter 7), we may write
ei=ej×ek
ei·(ej×ek),for the combination of subscriptsi, j, k=1, 2 ,3 and its cyclic permutations. A
similar expression holds foreiin terms of theei-basis. Moreover, it may be shown
that|e 1 ·(e 2 ×e 3 )|=
√
g.Show that the matrix[gij]is the inverse of the matrix[gij]. Hence calculate the con-
travariant componentsgijof the metric tensor in cylindrical polar coordinates.Using the index-lowering and index-raising properties ofgijandgijon an arbitrary vector
a, we find
δikak=ai=gijaj=gijgjkak.But, sinceais arbitrary, we must have
gijgjk=δki. (26.65)Denoting the matrix [gij]byGand [gij]byGˆ, equation (26.65) can be written in matrix
form asGGˆ =I,whereIis the unit matrix. HenceGandGˆare inverse matrices of each
other.