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Generalized covariant differentiation and axiom-based tensor analysis

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Abstract

This paper reports the new progresses in the axiomatization of tensor analysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentiations. These progresses strengthen the tendency of the axiomatization of tensor analysis.

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References

  1. Yin, Y. J., Chen, Y. Q., Ni, D., Shi, H. J., and Fan, Q. S. Shape equations and curvature bifurcations induced by inhomogeneous rigidities in cell membranes. Journal of Biomechanics, 38, 1433–1440 (2005)

    Article  Google Scholar 

  2. Yin, Y. J., Yin, J., and Ni, D. General mathematical frame for open or closed biomembranes: equilibrium theory and geometrically constraint equation. Journal of Mathematical Biology, 51, 403–413 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yin, Y. J., Yin, J., and Lv, C. J. Equilibrium theory in 2D Riemann manifold for heterogeneous biomembranes with arbitrary variational modes. Journal of Geometry and Physics, 58, 122–132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yin, Y. J. and Lv, C. J. Equilibrium theory and geometrical constraint equation for two-component lipid bilayer vesicles. Journal of Biological Physics, 34, 591–610 (2008)

    Article  Google Scholar 

  5. Yin, Y. J. and Wu, J. Y. Shape gradient: a driving force induced by space curvatures. International Journal of Nonlinear Sciences and Numerical Simulation, 11, 259–267 (2010)

    Article  MathSciNet  Google Scholar 

  6. Yin, Y. J., Chen, C., Lv, C. J., and Zheng, Q. S. Shape gradient and classical gradient of curva-tures: driving forces on micro/nano curved surfaces. Applied Mathematics and Mechanics (English Edition), 32(5), 533–550 (2011) DOI 10.1007/s10483-011-1436-6

    Article  MathSciNet  MATH  Google Scholar 

  7. Yin, Y. J. Extension of covariant derivative (I): from component form to objective form. Acta Mechanica Sinica, 31, 79–87 (2015)

    Article  MathSciNet  Google Scholar 

  8. Yin, Y. J. Extension of covariant derivative (II): from flat space to curved space. Acta Mechanica Sinica, 31, 88–95 (2015)

    Article  MathSciNet  Google Scholar 

  9. Yin, Y. J. Extension of the covariant derivative (III): from classical gradient to shape gradient. Acta Mechanica Sinica, 31, 96–103 (2015)

    Article  MathSciNet  Google Scholar 

  10. Ricci-Curbastro, G. Absolute differential calculus. Bulletin des Sciences Math matiques, 16, 167–189 (1892)

    Google Scholar 

  11. Ricci-Curbastro, G. and Levi-Civita, T. Methods of the absolute differential calculus and their applications. Mathematische Annalen, 54, 125–201 (1901)

    Article  Google Scholar 

  12. Huang, K. C., Xue, M. D., and Lu, M. W. Tensor Analysis, Tsinghua University Press, Beijing (2003)

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Yajun Yin

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  1. Yajun Yin
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Correspondence to Yajun Yin.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11072125 and 11272175), the Natural Science Foundation of Jiangsu Province (No. SBK201140044), and the Specialized Research Fund for Doctoral Program of Higher Education (No. 20130002110044)

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Yin, Y. Generalized covariant differentiation and axiom-based tensor analysis. Appl. Math. Mech.-Engl. Ed. 37, 379–394 (2016). https://doi.org/10.1007/s10483-016-2033-6

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  • DOI: https://doi.org/10.1007/s10483-016-2033-6

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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