Mathematical Methods for Physics and Engineering - Matematica.NET

26.15 THE METRIC TENSORsecond-order tensor T. Using the outer product notation in (26.23), we may writeT in three different ways:T = T ij e i ⊗ e j = T i je i ⊗ e j = T ij e i ⊗ e j ,where T ij , T i j and T ij are called the contravariant, mixed and covariant componentsof T respectively. It is important to remember that these three sets ofquantities form the components of the same tensor T but refer to different (tensor)bases made up from the basis vectors of the coordinate system. Again, if we areusing 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 contravariantand covariant components respectively; all others are called mixed components.26.15 The metric tensorAny particular curvilinear coordinate system is completely characterised at eachpoint in space by the nine quantitiesg ij = e i · e j , (26.56)which, as we will show, are the covariant components of a symmetric second-ordertensor g called the metric tensor.Since an infinitesimal vector displacement can be written as dr = du i e i , we findthat the square of the infinitesimal arc length (ds) 2 can be written in terms of themetric tensor as(ds) 2 = dr · dr = du i e i · du j e j = g ij du i du j . (26.57)It may further be shown that the volume element dV is given bydV = √ gdu 1 du 2 du 3 , (26.58)where g is the determinant of the matrix [ g ij ], which has the covariant componentsof 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 that e i · e j =0fori ≠ j) the metric tensor can be written in terms of thecoordinate-system scale factors h i , i =1, 2, 3as{h2g ij = i i = j,0 i ≠ j.Its determinant is then given by g = h 2 1 h2 2 h2 3 .957