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10.43. If A" are the components of an absolute contravariant tensor and A;A ) = 8, show that Aoi are the components of an absolute covariant tensor. The two tensors are said to be reciprocal. 10.44. If AU and Ay are the components of reciprocal symmetric tensors, and if x; are the components of a covariant vector, show that Ajaixi = Avixx; where *; = Atxa. 10.45. Show that the quantities egk = elk, where the elk are defined in Example 10.6(f), page 211, are the components of a covariant tensor of rank 3 and weight -1. 10.46. Let 611 = 0, 612 = Vg, 621 = -Vg, 622 = 0, where 9 = 911922 -(912)2. Show that the ey, i, j = 1, 2, are the components of a skew-symmetric covariant tensor such that z11 = 0, 612 = Vg, 621 = -Vg, 622 = 0. 10.47. Let el = egg giagib where cap is defined in the preceding problem. Show that ell = 0, 12 = 1/V/9, (21 = -1/Vg, 22 = 0. 10.48. Show that bi bej - b, be: = 0, i, j = 1, 2. 10.49. Prove that = du duB as aut aus + ; du du duy 10.50. Prove that duk = -gTak - gotTak. 10.51. Show that R112 = R221 = -R121 = -R212 = F- LN - M2 EG - F2 R212 = -R321 = GLN - M2 EG - F2 LN - M2 R121 = -Rizi = E FG - F2 and otherwise Rijk = 0. aTik 10.52. Prove that Ryk duck 10.53. Prove that 1 a log g 2 dul = ru + 12, 1 a log g 2 du2