Synopsis
In teaching the subject of tensor analysis, the authors were impressed by two characteristics of the available textbooks: the unnecessary narrowness of the subject, which is confined to quantities that transform in a few special ways, and the vagueness of the presentation, which leaves the student in doubt as to exactly how the quantities are manipulated. The paper attempts to broaden the subject and thus to carry on the pioneer work of Dr. Struik.
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References
John Wallis, letter to Collins, May 1673; Treatise on Algebra, London, 1685; F. Cajori, May 1673; Treatise on Algebra, London, 1685; F. Cajori, ‘Historical Notes on the Graphic Representation of Imaginaries Before the Time of Wessel’, Am. Math. Monthly 19 (1912) 167.
Gk. ’δλος = a whole (as in the English word holistic). See Moon and Spencer, Vectors, D. Van Nostrand Co., Princeton, N. J., 1965.
P. Moon and D. E. Spencer, Vectors, op. cit., pp. 7, 119; ‘The Meaning of the Vector Laplacian’, J. Franklin Inst. 256 (1953) 551.
Gk. Πλñθος. We are honestly trying to keep new words at a minimum, consistent with clarity. But ‘dimensionality’ or any other ordinary word is definitely ambiguous.
Here we follow J. A. Schouten and D. J. Struik, Einführung in die neuren Methoden der Differentialgeometrie, P. NoordhofF, Groningen, 1935.
From μέρος, a part. Note that merates are NOT components. The word ‘coordinates’ has been used, too, but it also is ambiguous. By ‘component’, we mean the scalar magnitude of the projection of a vector on an axis, not the component vector.
Even in such elementary questions, note the vagueness of the customary tensor treatment, where v i sometimes represents the holor and sometimes represents an arbitrary merate. See, for instance, J. L. Synge and A. Schild, Tensor calculus Univ. of Toronto Press, 1949; I. S. Sokolnikoff, Tensor Analysis, John Wiley and Sons, New York, 1964; B. Spain, Tensor Calculus Oliver and Boyd, Edinburgh, 1953.
F. D. Murnaghan, ‘The Generalized Kronecker Symbol’, Int. Math. Congress Proc., Univ. of Toronto Press, Toronto, 1924, p. 928; ‘The Generalized Kronecker Symbol and Its Application to the Theory of Determinants’, Am. Math. Monthly 32 (1925) 233; I. S. SokolnikofF, op. cit., p. 101.
P. Moon and D. E. Spencer, ‘A New Mathematical Representation of Alternating Currents’, Tensor 14 (1963) 110.
P. Moon and D. E. Spencer, Tensors, Chap. 5 (to be published). The name comes from the Greek άκίνητος = i invariant, fixed. Easily remembered from the English kinetic with negative prefix α-
Vector analysis can, of course, be extended to non-orthogonal coordinates, though ordinarily this is not done. See, for instance, our book Vectors, p. 207.
P. Moon and D. E. Spencer, Vectors, op. cit., p. 96.
Vectors, op. cit., p. 141.
J. W. Gibbs, ‘Elements of Vector Analysis’ in Scientific Papers of J. Willard Gibbs, Longmans, Green and Co., London, Vol. II, pp. 17–90; ‘On Multiple Algebra’, Am. Assn. for Adv. of Sei. Proc. 35 (1886) 37; E. B. Wilson, Vector Analysis, Yale Univ. Press, 1922.
P. Moon and D. E. Spencer, Field Theory for Engineers, D. Van Nostrand Co., Princeton, N. J., 1961; Field Theory Handbook, Springer-Verlag, Berlin, 1961.
For non-orthogonal coordinates, see Vectors, Equation (7.37).
P. Moon and D. E. Spencer, ‘The Meaning of the Vector Laplacian’, J. Franklin Inst. 256 (1953) 551; 258 (1954) 215; Also see Vectors, p. 235.
See, for instance, H. B. Phillips, Vector Analysis, John Wiley and Sons, New York, 1933.
Vectors, p. 323.
J. A. Schouten, Grundlagen der Vektor- und Affinor-Analysis, B. G. Teubner, Leipzig, 1914; D. J. Struik, Grundzüge der mehrdimensionalen Differentialgeometrie in direkter Darstellung, Springer-Verlag, Berlin, 1922; T. B. Drew, Handbook of Vector and Polyadic Analysis, Reinhold Pub. Corp., New York, 1961.
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© 1974 D. Reidel Publishing Company, Dordrecht-Holland
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Spencer, D.E., Moon, P. (1974). A Unified Approach to Hypernumbers. In: Cohen, R.S., Stachel, J.J., Wartofsky, M.W. (eds) For Dirk Struik. Boston Studies in the Philosophy of Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2115-9_9
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DOI: https://doi.org/10.1007/978-94-010-2115-9_9
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