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A short course on Tensor algebra and analysis, with applications to differential geometry

Paolo Vannucci

Abstract

This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text focuses on some subjects, with the intention of providing the reader with the main algebraic tools necessary for a modern course in continuum mechanics. The presentation of tensor algebra and analysis is intentionally done in Cartesian coordinates, that are normally used for classical problems. Then, an entire chapter is devoted to the passage to curvilinear coordinates and to the formalism of co- and contra-variant components. The tensor theory and results are specially applied to introduce some subjects concerning differential geometry of curves and surfaces. Also in this case, the presentation is mainly intended for applications to continuum mechanics.

A short course on Tensor algebra and analysis, with applications to differential geometry

Abstract

This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text focuses on some subjects, with the intention of providing the reader with the main algebraic tools necessary for a modern course in continuum mechanics. The presentation of tensor algebra and analysis is intentionally done in Cartesian coordinates, that are normally used for classical problems. Then, an entire chapter is devoted to the passage to curvilinear coordinates and to the formalism of co- and contra-variant components. The tensor theory and results are specially applied to introduce some subjects concerning differential geometry of curves and surfaces. Also in this case, the presentation is mainly intended for applications to continuum mechanics.
Paper Structure (82 sections, 861 equations, 38 figures)

This paper contains 82 sections, 861 equations, 38 figures.

Table of Contents

  1. Preface
  2. Acknowledgments
  3. List of symbols
  4. Points and vectors
  5. Points and vectors
  6. Scalar product, distance, orthogonality
  7. Basis of $\mathcal{V}$, expression of the scalar product
  8. Applied vectors
  9. Exercises
  10. Second rank tensors
  11. Second-rank tensors
  12. Dyads, tensor components
  13. Tensor product
  14. Transpose, symmetric and skew tensors
  15. Trace, scalar product of tensors
  16. ...and 67 more sections

Figures (38)

  • Figure 1: Same translation applied to two different points.
  • Figure 2: Sum of two vectors: the parallelogram rule.
  • Figure 3: Angle between two vectors.
  • Figure 4: Moment arm of an applied vector.
  • Figure 5: Scheme for the transposition of the moment.
  • ...and 33 more figures

Theorems & Definitions (49)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 39 more