in which is the Christoffel symbol of the second kind. Thus the quantities are deduced from the . Equations (67) and (70) are the foundation for the following discussion. Co-variant Differentiation of Tensors. If () is the vector resulting from an infinitesimal parallel displacement from to , and () the vector at the point , then the difference of these two, is also a vector. Since this is the case for an arbitrary choice of the , it follows that is a tensor, which we designate as the co-variant derivative of the tensor of the first rank (vector). Contracting this tensor, we obtain the divergence of the contra-variant tensor . In this we must observe that according to (70), [Pg 76] If we put, further, a quantity designated by Weyl as the contra-variant tensor density[16] of the first rank, it follows that, is a scalar density. [16]This expression is justified, in that has a tensor character. Every tensor, when multiplied by , changes into a tensor density. We employ capital Gothic letters for tensor densities. We get the law of parallel displacement for the co-variant vector by stipulating that the parallel displacement shall be effected in such a way that the scalar remains unchanged, and that therefore vanishes for every value assigned to (). We therefore get From this we arrive at the co-variant derivative of the co-variant vector by the same process as that which led to (71), [Pg 77] By interchanging the indices and , and subtracting, we get the skew-symmetrical tensor, For the co-variant differentiation of tensors of the second and higher ranks we may use the process by which (75) was deduced. Let, for example, () be a co-variant tensor of the second rank. Then is a scalar, if and are vectors. This expression must not be changed by the -displacement; expressing this by a formula, we get, using (67), , whence we get the desired co-variant derivative, In order that the general law of co-variant differentiation of tensors may be clearly seen, we shall write down two co-variant derivatives deduced in an analogous way: The general law of formation now becomes evident. From these formulae we shall deduce some others which are of interest for the physical applications of the theory. In case is skew-symmetrical, we obtain the tensor [Pg 78] which is skew-symmetrical in all pairs of indices, by cyclic interchange and addition. If, in (78), we replace by the fundamental tensor, , then the right-hand side vanishes identically; an analogous statement holds for (80) with respect to ; that is, the co-variant derivatives of the fundamental tensor vanish. That this must be so we see directly in the local system of co-ordinates. In case is skew-symmetrical, we obtain from (80), by contraction with respect to and , In the general case, from (79) and (80), by contraction with respect to and , we obtain the equations,