PL EN


Preferences help
enabled [disable] Abstract
Number of results
2019 | 137 | 96-118
Article title

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

Authors
Content
Title variants
Languages of publication
EN
Abstracts
EN
Extending the operating domain of 3D geometric differential operator expands also the range of its operations onto paired 3D dual reciprocal spaces as well as the scope of their validity into paired 3D multispatial structures. Intraspatial duality principle for paired 3D dual reciprocal spaces is inferred from differential operations performed on the dual reciprocal 3D spaces. The new scalar differential operator SCovar as multiplicative inverse of the scalar gradient differential operator SGrad is proposed here to deliver scalar components of covariant differentials in order to accommodate both the operational and structural legitimacy of differential and integral operations performed in 3D dual reciprocal spaces. From preliminarily formulated abstract intraspatial duality principle a generalized interspatial duality principle is deduced and the connection of paired multispatial structures established. It is shown that the finegrained geometric differential operator GDiff acts simultaneously in each of the paired dual reciprocal spaces, which is its formerly unknown operational feature.
Discipline
Year
Volume
137
Pages
96-118
Physical description
Contributors
author
  • Science/Mathematics Education Department, Southern University and A&M College, Baton Rouge, LA 70813, USA
References
  • [1] Tai C.-T. Generalized vector and dyadic analysis. Applied mathematics in field theory. Piscataway, NJ: IEEE Press, 1997, pp. 65, 73ff.
  • [2] Czajko J. Finegrained 3D differential operators hint at the inevitability of their dual reciprocaal portrayals. World Scientific News 132 (2019) 98-120
  • [3] Dieudonné J. Treatise on analysis III. New York: Academic Press, 1972, p. 303ff.
  • [4] Hestenes D. & Sobczyk G. Clifford algebra to geometric calculus: A unified language for mathematics and physics. Dordrecht: D. Reidel Publishing, 1987, p. 22ff.
  • [5] Hestenes D. A unified language for mathematics and physics. [pp. 1-23 in: Chisholm J.S.R. & Common A.K. (Eds.) Clifford Algebras and their Application to Mathematical Physics. Dordrecht: D. Reidel Publishing, 1986].
  • [6] Morris D. The quaternion Dirac equation. Port Mulgrave, UK: Abane & Right, 2016, p. 10f.
  • [7] Czajko J. Operational constraints on dimension of space imply both spacetime and timespace. Int. Lett. Chem. Phys. Astron. 36 (2014) 220-235
  • [8] Czajko J. Operational restrictions on morphing of quasi-geometric 4D physical spaces. Int. Lett. Chem. Phys. Astron. 41 (2015) 45-72
  • [9] 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.
  • [10] Czajko J. Quaternionic division by zero is implemented as multiplication by infinity in 4D hyperspace. World Scientific News 94(2) (2018) 190-216.
  • [11] Castonguay C. Meaning and existence in mathematics. New York: Springer, 1972, pp. 12f, 16, 22, 24f, 33, 46.
  • [12] Zwiebach B. A first course in string theory. Cambridge: Cambridge Univ. Press, 2005, p. 72.
  • [13] Essén M. Potential theory part I. [in: Aikawa H. & Essén M. Potential theory – selected topics. Berlin: Springer, 1996, see pp. 10, 16, 61].
  • [14] Aikawa H. Potential theory part II. [in: Aikawa H. & Essén M. Potential theory – selected topics. Berlin: Springer, 1996, see pp. 120, 114].
  • [15] Poincaré H. Théorie de potentiel Newtonien. Paris : Gauthier-Villars, 1899, p.4.
  • [16] Lumer G. Probleme de Cauchy et fonctions surharmoniques. [pp. 202-218 in: Hirsch F. & Mokobodzki G. (Eds.) Séminaire de Théorie du Potentiel, Paris, No. 2. Berlin: Springer, 1976].
  • [17] Lumer G. Probleme du Cauchy avec valeurs au bord continues, comportement asymptotique, et applications. [pp. 193-201 in: Hirsch F. & Mokobodzki G. (Eds.) Séminaire de Théorie du Potentiel, Paris, No. 2. Berlin: Springer, 1976].
  • [18] Helms L.L. Introduction to potential theory. New York: Wiley, 1969, pp. 223ff, 237f.
  • [19] Brelot M. Contributions to potential theory. Technical Note No. 2 KU-AF2-TN2. Lawrence, KS: Univ. of Kansas, 1955, pp. 13, 32, 88.
  • [20] Cipriami F. Dirichlet forms on noncommutative spaces. [pp.161-276 in: Schürmann M. & Franz U. (Eds.) Quantum potential theory. Berlin: Springer, see p. 245ff].
  • [21] Du Plessis N. An introduction to potential theory. Edinburgh: Oliver & Boyd, 1970, pp. 138ff, 39, 80f, 123ff.
  • [22] Blakely R.J. Potential theory in gravity and magnetic applications. Cambridge: Cambridge Univ. Press, 1996, p. 45.
  • [23] Kellog O.D. Foundations of potential theory. Berlin: Springer, 1929, p. 51ff.
  • [24] Heinonen J., Kilpeläinen T. & Martio O. Nonlinear potential theory of degenerate elliptic functions. Mineola, NY: Dover, 2006, p. 27ff.
  • [25] MacMillan W.D. Theoretical mechanics: The theory of potential. New York: Dover, 1958, p. 80ff.
  • [26] Lumer G. Probleme de Cauchy et fonctions surharmoniques. [pp.202-219 in : Brelot M., Choquet G. & Deny J. Séminaire de théorie du potentiel, Paris, No.2. Berlin : Springer, 1976].
  • [27] Pick M. Newton’s theory of potential. [pp. 19-36 in: Pick M., Picha J. & Vyskočil V. Theory of the Earth’s gravity field. Amsterdam: Elsevier Science Publishing Co., 1973].
  • [28] Borzeszkowski H.-H. von, Chrobok T. & Treder H.-J. Screening and absorption of gravitation in pre-relativistic and relativistic theories. [pp. 1-37 in: De Sabbata V., Gillies G.T. & Melnikov V.N. (Eds.) The gravitational constant: Generalized gravitational theories and experiments. Dordrecht: Kluwer, 2004, see p. 6].
  • [29] Balaji N. & Kumar S. Essential differential geometry. The language of general relativity. Printed in the USA, 2018, p. 40f.
  • [30] Tai C.T. A survey of the improper uses of  in vector analysis. Ann Arbor, MI: Univ. of Michigan Tech. Report RL 909, 1994.
  • [31] Tai C.T. A historical study of vector analysis. Ann Arbor, MI: Univ. of Michigan Tech. Report RL 915, 1995.
  • [32] Moon P. & Spencer D.E. Field theory handbook including coordinate systems, differential equations and their solutions. Berlin: Springer, 1971, pp. 3, 136f.
  • [33] Morris D. Lie groups and Lie algebras. A rewrite of Lie theory. Port Mulgrave: Abane & Right, 2016, p. 9.
  • [34] Czajko J. Path-independence of work done theorem is invalid in center-bound force fields. Stud. Math. Sci. 7(2) (2013) 25-39
  • [35] Czajko J. Wave-particle duality of a 6D wavicle in two paired 4D dual reciprocal quasispaces of a heterogeneous 8D quasispatial structure. World Scientific News 127(1) (2019) 1-55
  • [36] Hirsch M.W., Smale S. & Devaney R.L. Differential equations, dynamical systems, and an introduction to chaos. Amsterdam: Elsevier, 2016, p. 202f.
  • [37] O’Neil P.V. Advanced engineering mathematics. Pacific Grove, CA: Thompson, 2003, pp. 515, 518ff, 524.
  • [38] Stein S.K. & Barcellos A. Calculus and analytic geometry. New York: McGraw-Hill, 1992, pp. 816ff, 865ff.
  • [39] Stewart J. Calculus. Early transcendentals. Pacific Grove, CA: Brooks/Cole Publishing, 1999, pp. 215ff, 917ff.
  • [40] Swokowski E.W. Calculus. Late trigonometry version. Boston: PWS Publishing Company, 1992, p. 835ff.
  • [41] Larson R.E. & Hostetler R.P. Calculus with analytic geometry. Lexington, MA: D.C. Heath & Co., 1986, p. 803ff.
  • [42] Rogawski J. Calculus. New York: Freeman, 2008, p. 810ff.
  • [43] Thomas G.B., Jr. & Finney R.L. Calculus and analytic geometry II. Reading, MA: Addison-Wesley, 1996, p. 861.
  • [44] Halliday D., Resnick R. & Walker J. Fundamentals of physics volume 2: 8th edition. Hoboken, NJ: Wiley, 2008, p .664ff.
  • [45] Feynman R.P., Leighton R.B. & Sands M. The Feynman lectures on physics 2. Mainly electromagnetism and matter. Reading, MA: Addison-Wesley, 1977, p. 6-11f.
  • [46] Purcell E.M. Electricity and magnetism. Berkeley physics course volume 2e: in SI units. Chennai: McGraw-Hill, 2011, pp. 97ff, 103ff.
  • [47] Jefimenko O.D. Electricity and magnetism. An introduction to the theory of electric and magnetic fields. Star City, WV: Electret Scientific Company, 1989, p. 116f.
  • [48] Constant F.W. Theoretical physics. Mechanics of particles, rigid and elastic bodies, fluids and heat flow. Huntington, NY: Robert E. Krieger Publishing, 1978, p. 82ff.
  • [49] Tribble A.C. Princeton guide to advanced physics. Princeton, NJ: Princeton Univ. Press, 1996, pp. 65, 128ff.
  • [50] Landau L.D. & Lifschitz E.M. Mechanics. Amsterdam: Elsevier, 2016, pp. 8ff, 13, 15, 108ff.
  • [51] Friedman B. Principles and techniques of applied mathematics. New York: Wiley, 1990, p. 144f.
  • [52] Arfken G. & Weber H.J. Mathematical methods for physicists. Amsterdam: Elsevier, 2005, p. 177.
  • [53] Dunford N. & Schwartz J.T. Linear operators. Part 1: General theory. New York: Interscience, 1967, p. 36.
  • [54] Czajko J. On Conjugate Complex Time I: Complex Time Implies Existence of Tangential Potential that Can Cause Some Equipotential Effects of Gravity. Chaos Solit. Fract. 11 (2000) 1983-1992
  • [55] Czajko J. On Conjugate Complex Time II: Equipotential Effect of Gravity Retrodicts Differential and Predicts Apparent Anomalous Rotation of the Sun. Chaos Solit. Fract. 11 (2000) 2001-2016
  • [56] Dunford N. & Schwartz J.T. Linear operators. Part 3: Spectral operators. New York: Interscience, 1972, pp. 2434ff.
  • [57] Sommerfeld A. Lectures on theoretical physics. II: Mechanics of deformable bodies. New York: Academic Press, 1956, p. 88f.
  • [58] Tapp K. Differential geometry of curves and surfaces. Switzerland: Springer, 2016, p. 14ff.
  • [59] Oprea J. Differential geometry and its applications. Washington, DC: MAA, 2007, p. 14ff.
  • [60] Banchoff T. & Lovett S. Differential geometry of curves and surfaces. Natick, MA: A.K. Peters, 2010, p. 87ff.
  • [61] Czajko J. On unconventional division by zero. World Scientific News 99 (2018) 133-147
  • [62] Norton R.E. Complex variables for scientists and engineers. An introduction. New York: Oxford Univ. Press, 2010, p. 294f.
  • [63] Okubo S. Introduction to octonion and other non-associative algebras in physics. Cambridge: Cambridge Univ. Press, 1995, p. 14ff.
  • [64] Dray T. & Manogue C.A. The geometry of the octonions. Singapore: World Scientific, 2015, p. 18f.
Document Type
article
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.psjd-8ec1629f-0205-458c-9f10-d6d609718ce2
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.