THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Xiao-Jun Yang

THE VECTOR CALCULUS WITH RESPECT TO MONOTONE FUNCTIONS APPLIED TO HEAT CONDUCTION PROBLEMS

ABSTRACT
This paper addresses the theory of the vector calculus with respect to monotone functions for the first time. The Green-like theorem, Stokes-like theorem, Gauss-like theorem, and Green-like identities are obtained with the aid of the notation of Gibbs. The results are used to model the heat-conduction problems arising in the complex phenomenon.
KEYWORDS
vector calculus with respect to monotone functions, Green-like theorem, Stokes-like theorem, Gauss-like theorem, Green-like identities, heat-conduction problems
PAPER SUBMITTED: 2019-08-18
PAPER REVISED: 2020-01-20
PAPER ACCEPTED: 2020-01-25
PUBLISHED ONLINE: 2020-11-27
DOI REFERENCE: https://doi.org/10.2298/TSCI2006949Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Issue 6, PAGES [3949 - 3959]
REFERENCES
  1. Marsden, J. E., Tromba, A., Vector Calculus, Macmillan, London, UK, 2003
  2. Maxwell, J. C., A treatise on electricity and magnetism, Clarendon Press, Oxford, UK, 1873
  3. Eddington, A. S., The Mathematical Theory of Relativity, The Cambridge University Press, Cambridge, UK, 1923
  4. Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, The Cambridge University Press, Cambridge, UK, 1906
  5. Carslaw, H. S., Introduction to the Mathematical Theory of the Conduction of Heat in Solids, Dover Publication, Mineola, N. Y., USA, 1906
  6. Gibbs, J. W., Elements of Vector Analysis, New Haven, Nashville, Tenn., USA, 1881
  7. Gibbs, J. W. The Scientific Papers of J. Willard Gibbs, Longmans, Green and Company, London, UK, 1906
  8. Heaviside, O., Electromagnetic Theory. The Cambridge University Press, Cambridge, UK, 1893 (Reprinted by Cosimo in 2008)
  9. Gauss C. F., Theoria Attractionis Corporum Sphaeroidicorum Ellipticorum Homogeneorum Methodo Novo Tractata, Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 2 (1813), pp. 2-5
  10. Green, G., An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism, Notingham, London, UK, 1828
  11. Katz, V. J., The History of Stokes' Theorem, Mathematics Magazine, 52 (1979), 3, pp. 146-156
  12. Ostrogradsky, M. V., Note sur la théorie de la chaleur, Mémoires présentés à l'Académie impériale des Sciences de St. Petersbourg, 6 (1831), 1, pp. 123-138 (Presented in 1828)
  13. Ostrogradsky, M. V., Mémoire sur l'equilibre et le mouvement des corps elastiques, Mémoires présentés à l'Académie impériale des Sciences de St. Petersbourg, 6 (1831), 1, pp. 39-53 (Presented in 1828)
  14. Stokes, G. G., A Smith's Prize Paper, The Cambridge University Press, Cambridge, UK, 1854
  15. Hankel, H. Zur allgemeinen Theorie der Bewegung der Flussigkeiten, Dieterische University, Buchdruckerei, 1861
  16. Stephenson, R. J., Development of Vector Analysis from Quaternions, American Journal of Physics, 34 (1966), 3, pp. 194-201
  17. Gibbs, J. W., Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Yale University Press, New Haven, Nashville, Tenn., USA, 1901
  18. Taylor, A., Mann, W. R., Advanced Calculus, 2nd ed., Xerox, Norwalk, Conn., USA, 1972
  19. Leibniz, G. W., Memoir Using the Chain Rule, 1676
  20. Stieltjes, T. J., Recherches Sur les Fractions Continues, Comptes Rendus de l'Académie des Sciences Series I - Mathematics, 118 (1894), pp. 1401-1403
  21. Riemann, B. Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Dieterich, Göttingen, 1867
  22. Yang, X.-J., Theory and Applications of Special Functions for Scientists and Engineers, Springer Nature, New York, USA, 2021
  23. Yang, X.-J., et al., General Fractional Derivatives with Applications in Viscoelasticity, Academic Press, New York, USA, 2020
  24. Yang, X. J., New Non-conventional Methods for Quantitative Concepts of Anomalous Rheology, Thermal Science, 23 (2019), 6B, pp. 4117-4127
  25. Widder, D. V., Advanced Calculus, Prentice-Hall, New York, USA, 1947
  26. Richard, E., et al. Calculus of Vector Functions, Prentice-Hall, Upper Saddle River, N. Y., USA, 1968
  27. Stolze, C. H., A History of the Divergence Theorem, Historia Mathematica, 5 (1978), 4, pp. 437-442
  28. Fourier, J. B. J., Theorie Analytic da la Chaleur, Paris, 1822 (Translated 1878)
  29. Laplace, P. S., Théorie des Attractions des Sphéroïdes et de la Figure des Planètes, Mémoires de l'Académie Royale des Sciences, (1782), pp. 113-196
  30. Poisson, S. D. Mémoire sur l'équilibre et le Mouvement des Corps élastiques, Didot, Paris, France, 1813
PDF VERSION [DOWNLOAD]
The vector calculus with respect to monotone functions applied to heat conduction problems

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence