Tensor calculus (Book)

Tensorial calculation. This book addresses the importance of tensor calculus in mathematical and physical culture , providing students with the means to approach the study of the great theories of contemporary physics.

Foreword

In 1900 , in a memory that has become famous, Ricci and Levi CivitaThey published the first systematic replacement regarding tensor calculus, drawing the attention of mathematicians and physicists to a number of its possible applications. Since then, the road traveled has been long. The emergence of the theory of relativity, which has only been possible thanks to the prior existence of tensor calculus, has in turn made him experience enormous progress. In this way, these computational techniques have become one of the most powerful instruments in all modern theoretical physics. Recently, they have even been used in the study of technical problems such as the interconnection of electrical machines. It is possible to say that the tensorial calculation will have to comprise in the future of any mathematical or physical culture.

This short treatise has been divided into two parts: one relating to algebra and tensor analysis, and the other to its most important applications. In the first of these, tensor algebra has been completed with some pages specially devoted to external algebra, the knowledge of which is of particular interest to physicists. On the contrary, concepts such as tensor density and tensor capacity have been avoided, the current mathematical interest of which seems very limited. The notion of an attached tensioner of an antisymmetric tensioner, moreover, makes it possible to supply such an omission sufficiently.

As regards tensor analysis, we have intentionally limited ourselves to exposing the analysis of the tensor fields in a Riemann space , as Riemannian geometry is the one of greatest interest from the point of view of applications. Elie Cartan’s mobile reference system method has been systematically adopted , which, being the most geometric and intuitive, also offers the advantage of allowing the reader to undertake the study of other generalized geometries without great effort.

In the application section, it was naturally necessary to make a selection. A first chapter is intended to show how the geometry of Riemann spaces becomes intuitive as soon as it comes into contact with classical analytical dynamics, and the help it is able to bring to it. In particular, we have dedicated an introduction to the study of continuous media and elasticity. The reader willing to delve into these matters can consult the works of Léon Brillouin .

Two other chapters are devoted to the study of equations of Maxwell of the electromagnetism and the theory of relativity . As regards the relativistic theory of gravitation , of which only its principles have been outlined, the reader will be able to consult the interesting work of Georges Darmois .

As an author, I will be satisfied that I have fulfilled my purpose if I have provided students with the means to approach the study of the great theories of contemporary physics.

Content

  1. PART I: Tensor calculation.
  2. Vector spaces.
  3. The related and Euclidean point spaces.
  4. Tensor algebra.
  5. Euclidean space in curvilinear coordinates.
  6. Riemannian spaces.
  7. PART II: Applications.
  8. Tensor calculus and classical dynamics.
  9. The theory of restricted relativity and Maxwell’s equations.
  10. Elements of the relativistic theory of gravitation.

 

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