Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 201In such a case we say that the contravariant vector u' and covariant vector u, areassociated. We regard the u' and u, as different aspects of one underlying geometricobject u, a vector field abstracted from its components. Thus we speak of u' as contravariantcomponents of u, u, as covariant components of u. In standard coordinatesthe two types of components coincide and can be identified with our usual vectorcomponents.The contravariant and covariant vectors are the tensors of Jirst order. We alsointroduce tensors of order zero, as scalar functions whose values are not changed bycoordinate transformations. Thus in the two coordinate systems (x') and (Ti), suchrespectively, wherea function is given by f (xl,..., xn) and ?(TI, ..., in),f (XI,.. ., x") = f(21, ..., 5")whenever (xi) and (f ') refer to the same point; that is,We also call such a tensor an invariant.We can also introduce tensors of higher order. A tensor of order two requires twoindices and hence can be arranged as a matrix. For example, we denote by v;, a tensorof order two that is covariant in both indices; here vi, (i = 1 , . . . , n, j = 1 , . . . , n)are the n2 components of the tensor in the (xi) coordinate system. In the secondcoordinate system (2') the tensor has components Eii, and we require thatBecause of the many such sums appearing in tensor analysis, one agrees to drop thesigma signs and to write (3.79) simply aswith the understanding that we sum over each index which appears more than once(k and 1 in this case). This notational rule is called the summation convention.By vf we denote a second-order tensor, called mixed, which is contravariant inthe index i and covariant in the index j, and require that for a change of coordinates,By v'j we denote a second-order tensor that is contravariant in both indices andrequire thatSmilar definitions are given for tensors of third, fourth, and higher orders. Forexample, wy denotes a tensor contravariant in i and j and covariant in k, whereby