Generalized covariant differentiation and axiom-based tensor analysis

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• 作者：Yajun Yin
• 关键词：tensor analysis ; axiom of covariant form invariability ; generalized component ; generalized covariant differentiation ; covariant differential transformation group
• 刊名：Applied Mathematics and Mechanics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：37
• 期：3
• 页码：379-394
• 全文大小：169 KB
• 参考文献：[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)CrossRef
  [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)MathSciNet CrossRef MATH
  [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)MathSciNet CrossRef MATH
  [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)CrossRef
  [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)MathSciNet CrossRef
  [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/s10483-011-1436-6MathSciNet CrossRef MATH
  [7]Yin, Y. J. Extension of covariant derivative (I): from component form to objective form. Acta Mechanica Sinica, 31, 79–87 (2015)MathSciNet CrossRef
  [8]Yin, Y. J. Extension of covariant derivative (II): from flat space to curved space. Acta Mechanica Sinica, 31, 88–95 (2015)MathSciNet CrossRef
  [9]Yin, Y. J. Extension of the covariant derivative (III): from classical gradient to shape gradient. Acta Mechanica Sinica, 31, 96–103 (2015)MathSciNet CrossRef
  [10]Ricci-Curbastro, G. Absolute differential calculus. Bulletin des Sciences Math matiques, 16, 167–189 (1892)
  [11]Ricci-Curbastro, G. and Levi-Civita, T. Methods of the absolute differential calculus and their applications. Mathematische Annalen, 54, 125–201 (1901)CrossRef
  [12]Huang, K. C., Xue, M. D., and Lu, M. W. Tensor Analysis, Tsinghua University Press, Beijing (2003)
  
• 作者单位：Yajun Yin (1)
  
  1. Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Applications of Mathematics
  Mechanics
  Mathematical Modeling and IndustrialMathematics
  Chinese Library of Science
  
• 出版者：Shanghai University, in co-publication with Springer
• ISSN：1573-2754

文摘

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.