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On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations


Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general four-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored.

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  1. 1.

    L. Euler (1755) Hist. de l’Acad. de Berlin 11 274–315

    Google Scholar 

  2. 2.

    M. Kline (1980) Mathematics: The Loss of Certainty Oxford University Press New York

    Google Scholar 

  3. 3.

    M. Kline (1972) Mathematical Thought from Ancient to Modern Times NumberInSeriesVol. 2 Oxford University Press New York

    Google Scholar 

  4. 4.

    R. E. Meyer, Introduction to Mathematical Fluid Dynamics (Wiley, 1972).

  5. 5.

    B. Dubrovin, S. Novikov, and A. Fomenko, Modern Geometry, Vol. 1 (Ed. Mir, Moscow, 1982).

  6. 6.

    A.E. Chubykalo R. Smirnov-Rueda (1997) Mod. Phys. Lett. A 12 IssueID1 1 Occurrence Handle97i:78005

    MathSciNet  Google Scholar 

  7. 7.

    A. E. Chubykalo, R. A. Flores, and J. A. Perez, Proceedings of the International Congress, ‘Lorentz Group, CPT and Neutrino’, Zacatecas University (Mexico, 1997) pp. 384.

  8. 8.

    A.E. Chubykalo R. Alvarado-Flores (2002) Hadronic J. 25 159 Occurrence Handle2003f:26013

    MathSciNet  Google Scholar 

  9. 9.

    A. Chubykalo A. Espinoza R. Flores-Alvarado (2004) Hadronic J. 27 IssueID6 625

    Google Scholar 

  10. 10.

    G.K. Batchelor (1967) Introduction to Fluid Dynamics Cambridge University Press Cambridge

    Google Scholar 

  11. 11.

    L.D. Landau E.M. Lifshitz (1973) Classical Theory of Fields Nauka Moscow

    Google Scholar 

  12. 12.

    A.E. Chubykalo R. Smirnov-Rueda (1996) Phys. Rev. E 53 IssueID5 5373 Occurrence Handle10.1103/PhysRevE.53.5373 Occurrence Handle97b:78007

    Article  MathSciNet  Google Scholar 

  13. 13.

    A.E. Chubykalo R. Smirnov-Rueda (1998) Phys. Rev. E 57 IssueID3 3683 Occurrence Handle10.1103/PhysRevE.57.3683

    Article  Google Scholar 

  14. 14.

    R. Smirnov-Rueda (2005) Found. Phys. 35 IssueID1 1 Occurrence Handle10.1007/s10701-004-1911-z Occurrence Handle2133909

    Article  MathSciNet  Google Scholar 

  15. 15.

    A. Chubykalo A. Espinoza V. Onoochin R. Smirnov-Rueda (Eds) (2004) Has the Last Word Been Said on Classical Electrodynamics? New Horizons Rinton Press Princeton

    Google Scholar 

  16. 16.

    V.I. Arnold (1974) Mathematical Methods of Classical Mechanics Nauka Moscow

    Google Scholar 

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Smirnov-Rueda, R. On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations. Found Phys 35, 1695–1723 (2005).

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  • final Cauchy problem
  • continuity equation
  • convection theorem
  • fluid quantity
  • Maxwell’s equations