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2020 | 144 | 358-371
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New product differentiation rule for paired scalar reciprocal functions

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When integral kernel of an integral transform is being formed, it should be the outcome of scalar product differentiation rule if the kernel is supposed to be eventually used as an integrand in a prospective integration. Yet it has already been shown that despite ensuing from properly performed differentiation, the resulting integral kernel contains, beside the covariant differential that is suitable for integration, also a certain contravariant term, which is not appropriate for integration in the same space as the covariant differential. But the contravariant term also can be turned into proper, though multiplicatively inverse covariant differential, if placed within a space that is reciprocal to the given primary space in which the first, covariant differential, is represented naturally. This uncharacteristic conversion of the contravariant expression from the primary space into the reciprocal covariant differential in the dual reciprocal space that is paired with the given primary space, can be considered as indirect proof that pairing of mutually dual reciprocal spaces is necessary in order to properly form operationally legitimate and geometrically valid differential structures. Consequently, the pairing of an infinitesimal descending singularity of the 2D domain of complex numbers with an infinitely ascending singularity deployed in the 1D domain of real numbers requires certain dual reciprocal spatial or quasispatial structures, for the downward transition from 2D descending complex singularities to the 1D ascending “real” singularities to be meaningfully/unambiguously implemented. Furthermore, just as integration by parts formula is a counterpart of the regular product differentiation rule, a new multispatial scalar product integration rule is proposed as a counterpart to the singlespatial product differentiation rule, and introduced by analogy to the latter, “regular” product integration rule.
Physical description
  • Science/Mathematics Education Department, Southern University and A&M College, Baton Rouge, LA 70813, USA
  • [1] Czajko J. Pairing of infinitesimal descending complex singularity with infinitely ascending, real domain singularity. World Scientific News144 (2020) 56-69
  • [2] Cheng E. Beyond infinity. An expedition to the outer limits of mathematics. New York: Basic Books, 2017, p. 13ff.
  • [3] Rudin W. Real and complex analysis. New Delhi: McGraw-Hill, 2006, p.18f.
  • [4] Maurin K. Analysis II: Integration, distributions, holomorphic functions, tensor and harmonic analysis. Dordrecht: Reidel, 1980, pp. 65, 726.
  • [5] Stewart I. & Tall D. Complex analysis. (The hitchhiker’s guide to the plane). Cambridge: Cambridge Univ. Press, 1996, p. 224f.
  • [6] Norton R.E. Complex analysis for scientists and engineers. An introduction. Oxford: Oxford Univ. Press, 2010, p. 159f.
  • [7] Boas R.P. Invitation to complex analysis. Washington, DC: MAA, 2010, p. 3.
  • [8] Stein S.K. & Barcellos A. Calculus and analytic geometry. New York: McGraw-Hill Book, 1992, pp. 87, 134, 228.
  • [9] Jeffrey A. Handbook of mathematical formulas and integrals. San Diego: Academic Press, 2000, p.87.
  • [10] Stroud K.A. & Booth D. Advanced engineering mathematics. New York: Industrial Press, 2003, p. 275.
  • [11] Larson R.E. & Hostetler R.P. Calculus with analytic geometry. Lexington, MA: D.C. Heath & Co., 1986, p. 322ff.
  • [12] Stewart J. Calculus. Early transcendentals. Pacific Grove, CA: Brooks/Cole Publishing, 1999, p. 470.
  • [13] Poisson S.D. Traité de la mécanique. Bruxelles : Hauman et Compagnie, 1838, p. 133.
  • [14] Gellert W. et al (Eds.), The VNR concise encyclopedia of mathematics. Second edition. New York: Van Nostrand Reinhold, 1989, p. 704f.
  • [15] Madsen I. & Tornehave J. From calculus to cohomology. De Rham cohomology and characteristic classes. Cambridge: Cambridge Univ. Press, 1998, p. 59.
  • [16] Appel W. Mathematics for physics and physicists. Princeton, NJ: Princeton Univ. Press, 2007, p. 423.
  • [17] Iyanaga S. & Kawada Y. (Eds.) Encyclopedic dictionary of mathematics A-M. Cambridge, MA: The MIT Press, 1977, p. 698ff.
  • [18] Rodnianski I. The heat equation. [pp. 217-219 in: Gowers T., Barrow-Green J. & Leader I. (Eds.) The Princeton companion to mathematics. Princeton, NJ: Princeton Univ. Press, 2008, see p. 218f].
  • [19] John F. A walk through partial differential equations. [p.73-84 in: Chern S.S. (Ed.) Seminar on nonlinear differential equations. New York: Springer, 1984, see pp.73, 80, 82].
  • [20] Czajko J. Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces. World Scientific News 92(2) (2018) 171-197.
  • [21] Czajko J. Quaternionic division by zero is implemented as multiplication by infinity in 4D hyperspace. World Scientific News 94(2) (2018) 190-216.
  • [22] Vladimirov V.S. Methods of the theory of functions of many complex variables. Mineola, NY: Dover, 2007, p. 288ff.
  • [23] Morrow J. & Kodaira K. Complex manifolds. New York: Holt, Rinehart and Winston, 1971, p.10.
  • [24] Lichnerowicz A. Linear algebra and analysis. San Francisco: Holden-Day, 1967, pp. 256ff, 260, 275, 279.
  • [25] Czajko J. Finegrained 3D differential operators hint at the inevitability of their dual reciprocal portrayals. World Scientific News 132 (2019) 98-120.
  • [26] Czajko J. Dual reciprocal scalar potentials paired via differential operators in Frenet frames make the operators to act simultaneously in each of two paired 3D reciprocal spaces. World Scientific News 137 (2019) 96-118
  • [27] Czajko J. Mathematical gateway to complementary hidden variables in macrophysics. International Letters of Chemistry, Physics and Astronomy 50 (2015) 117-142
  • [28] Morley F. & Morley F.V. Inversive geometry. New York: Chelsea Publishing, 1954, pp. 39, 44, 74.
  • [29] Nahin P.J. Inside interesting integrals (with an introduction to contour integration). New York: Springer, 2015, pp. 14f, 15, 16.
  • [30] Czajko J. Operational constraints on dimension of space imply both spacetime and timespace. International Letters of Chemistry, Physics and Astron 36 (2014) 220-235
  • [31] Czajko J. Operational restrictions on morphing of quasi-geometric 4D physical spaces. International Letters of Chemistry, Physics and Astronomy 41 (2015) 45-72
  • [32] Thomas G.B., Jr. & Finney R.L. Calculus and analytic geometry II. Reading, MA: Addison-Wesley, 1996, p. 998.
  • [33] Tartar L. Compacité par compensation : résultats et perspectives. [p. 350-369 in: Brezis H. & Lions J.L. (Eds.) Nonlinear partial differential equations and their applications. Collège de France Seminar IV. Boston: Pitman, 1983, see p. 364].
  • [34] Silvestrov P.G. Localization in imaginary vector potential. Physical Review B (Condensed Matter and Materials Physics) 58(16) (1998) R10111-R10114
  • [35] Robinson A. Non-standard analysis. Princeton, NJ: Princeton Univ. Press, 1996, pp. 261f, 267, 269, 276, 282.
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