Mathematical Methods for Physics and Engineering - Matematica.NET

26.17 RELATIVE TENSORS◮Show that the quantities g ij = e i · e jtensor.form the covariant components of a second-orderIn the new (primed) coordinate system we haveg ij ′ = e ′ i · e ′ j,but using (26.67) for the inverse transformation, we havee ′ i = ∂uk∂u e k,′iand similarly for e ′ j .Thuswemaywriteg ij ′ = ∂uk ∂u le∂u ′i ∂u ′j k · e l = ∂uk ∂u l∂u ′i ∂u g kl,′jwhich shows that the g ij are indeed the covariant components of a second-order tensor(themetrictensorg). ◭A similar argument to that used in the above example shows that the quantitiesg ij form the contravariant components of a second-order tensor which transformsaccording tog ′ ij = ∂u′ i∂u ′ j∂u k ∂u l gkl .In the previous section we discussed the use of the components g ij and g ij inthe raising and lowering of indices in contravariant and covariant vectors. Thiscan be extended to tensors of arbitrary rank. In general, contraction of a tensorwith g ij will convert the contracted index from being contravariant (superscript)to covariant (subscript), i.e. it is lowered. This can be repeated for as many indicesare required. For example,Similarly contraction with g ij raises an index, i.e.T ij = g ik T k j = g ik g jl T kl . (26.72)T ij = g ik T jk = gik g jl T kl . (26.73)That (26.72) and (26.73) are mutually consistent may be shown by using the factthat g ik g kj = δ i j .26.17 Relative tensorsIn section 26.10 we introduced the concept of pseudotensors in the context of therotation (proper or improper) of a set of Cartesian axes. Generalising to arbitrarycoordinate transformations leads to the notion of a relative tensor.For an arbitrary coordinate transformation from one general coordinate system963