Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 203and hence I.- ( atr axk ) (atr ax1) - axk ax1 atr atrg..- -- -- - ----axk ax1 ax' afj a?; a3F.i axk ax'axk ax1---a ~ an) i gk'.Therefore gij is indeed a second-order tensor, covariant in both indices. We call gljthe fundamental metric tensor.We observe that the g,, reduce to (Kronecker delta) in standard coordinates,since in the (t'),Hence we can regard the gij as the covariant tensor obtained from Sij (constantfunctions) in standard coordinates. From these functions we obtain six associatedtensors as earlier. However, since Sji = Si,, gji = gij (as noted before), so that thesetwo associated tensors are the same. By setting G) = S;, in standard coordinates (ti)we obtain a mixed second-order tensor g;. In the (x') coordinates we obtain(since the matrices (axi/atj) and (ati/axj) are inverses of each other). Henceg! = g! = Sij (constant functions)J 1 (3.85)in all coordinates. We often write 6) for this tensor. Next, by setting G'j = Sij instandard coordinates we obtain a contravariant second-order tensor giJ; as earlier,Comparison with (3.84) suggests that (giJ) is the inverse matrix of (gij), that is, thatThis can be directly verified (Problem 6). It follows that both matrices are nonsingular.We denote by g the determinant of (glj):g = det (gij). (3.88)Thus g is a scalar function, depending on the coordinate system; it is not an invariant.One can show that g must be positive (Problem 7).We thus see that our fundamental metric tensor has three aspects:gij = gj; , gf = gl = S!, g" = gii (inverse matrix of the g;,). (3.89)Tensor algebra. Tensors can be combined in certain ways to yield new tensors.We can add two tensors of the same type, to obtain another of the same type (samenumber of contravariant and covariant indices)-for example,