Advanced Calculus fi..

210 Advanced Calculus, Fifth Editionand 0. It is left as an exercise to verify that the new objects are indeed alternatingtensors (Problem 22 following Section 3.1 1).With the aid of the operators A we can now obtain an alternating product of twoalternating tensors. We can simply apply A to the tensor product of the two tensors.The result, slightly modified by a constant factor, is called the exterior product (orwedge product) of the two alternating tensors: If uil ,,i, and vjl ,,r are alternatingtensors, the exterior product is written as uil,,,;, A v,, ,,js and is defined asThus ui A vjk equalsThis is a new alternating tensor w;jk of order 3.For more information on alternating tensors and references, see Sections 5.19to 5.21.PROBLEMS1. In E~ let (el, 5') be standard coordinates and let (xl, x2) be new coordinates given byx1 = 35' + 2t2, x2 = 4t1 + 36'. Find the (xi) components of the following tensors,for which the components in standard coordinates are given.a) u;, where U1 = 5'e2, U2 = 6' - 5'.b) v', where V' = 6' cost2, v2 = 6' sint2.C) w;j, where Wll = 0, W12 = (It2, W2] = WZZ = 0.d) zi., where Z: = 6' + c2, Z: = Z: = 35' + 2t2, 2: = 6' - {2.I2. For the coordinates of Problem 1, find (a) the components of the fundamental metrictensor glj in standard coordinates and in the (xi) and (b) the gj and g'j in both sets ofcoordinates.3. With reference to the coordinates and tensors of Problem 1 and with the aid of the resultsof Problem 2, find the following tensor components in (xi):a) ui, associated to ui; b) vi , associated to vi ; c) wij, associated to wi,.4. a) Show that if a tensor has all components 0 at a point in one coordinate system, thenall components are 0 at the point in every allowed coordinate system.b) Show that if two tensors of the same type have corresponding components equal(identically) in one coordinate system, then cooresponding components are equal inevery allowed coordinate system. (One then says that the tensors are equal.)C) Show that if a tensor uij is such that u;, =-uj; in one coordinate system (xi), theniiij = P j; in every other coordinate system (xl). (One then calls the tensor symmetric.)d) Show that if a tensor uij is such that ui, = -uj;, in one coordinate system (xi), theniiij = -ii,i in every other coordinate system (P' ).(One calls such a tensor alternating, as in Section 3.11.)5. Prove: If components U; are defined in (ti), and corresponding components u; are definedin every other allowed coordinate system (xi) by the equation u; = (a6j/axi)Uj, thenthe components iii in (.ti) satisfy (3.73, so that a covariant vector has been defined.