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Intrinsic Vector and Tensor Techniques in Minkowski Space with Applications to Special Relativity
1.See, e.g., J. W. Gibbs and E. B. Wilson, Vector Analysis (Dover Publications, Inc., New York, 1960);
1.A. P. Wills, Vector Analysis with an Introduction to Tensor Analysis (Dover Publications, Inc., New York, 1958);
1.C. E. Weatherburn, Advanced Vector Analysis with Applications to Mathematical Physics (G. Bell and Sons, London, 1944);
1.T. B. Drew, Handbook of Vector and Polyadic Analysis (Reinhold Publishing Corporation, New York, 1961);
1.L. Brand, Vector and Tensor Analysis (John Wiley & Sons, Inc., New York, 1947);
1.P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill Book Company, Inc., New York, 1953).
2.S. Chapman and T. G. Cowling, The Mathematical Theory of Non‐Uniform Gases (Cambridge University Press, Cambridge, 1960).
3.However, vectors in flat Riemann spaces are sometimes represented intrinsically. See, e.g., A. Lichnerowicz, Elements of Tensor Calculus (Methuen and Company Ltd., London, 1962);
3.D. E. Rutherford, Vector Methods (Oliver and Boyd, Edinburgh, 1954).
4.L. L. Foldy, J. Math. Phys. 6, 1871 (1965).
5.H. E. Moses, Nuovo Cimento Suppl. 7, 1 (1958).
6.J. S. Lomont, Phys. Rev. 111, 1710 (1958).
7.B. Kurşunoğlu, J. Math. Phys. 2, 22 (1961).
8.A. J. Macfarlane, J. Math. Phys. 3, 1116 (1962).
9.In the following discussion we use underlined boldface type to denote four‐vectors in Minkowski space in order to distinguish them from vectors in Euclidean space (for which the conventional boldface roman type is used). Latin indices will take on the values 1, 2, 3, and Greek indices will take on the values 0, 1, 2, 3. Unless otherwise stated, the summation convention will apply to repeated indices in any expression.
10.Dyadics will be denoted by sans serif type, while boldface Greek capitals are used for the few higher‐order polyadics which appear in the text.
11.For example, see Lichnerowicz, Ref. 3; also H. Flanders, Differential Forms (Academic Press Inc., New York, 1963).
12.S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row and Peterson, Evanston, Illinois, 1961), p. 40.
13.For a comparison of the dyadic forms obtained below with their corresponding component representations see, for example, L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Ltd., Oxford, England, 1962), Chap. 4;
13.V. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, Ltd., Oxford, England, 1964), Chap. 2. Also see Ref. 12, Sec. 7d.
14.H. S. Ruse, Proc. London Math. Soc. 41, 302 (1936).
15.The power of a dyadic will be defined by and
16.The commutator of two dyadics is defined by
17.Other polynomial forms for finite restricted homogeneous Lorentz transformations are discussed by S. L. Bazanski, J. Math. Phys. 6, 1201 (1965).
18.E. Wigner, Ann. Math. 40, 149 (1939).
19.B. Friedman, Principles and Techniques of Applied Mathematics (John Wiley & Sons, Inc., New York, 1956).
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