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

TL;DR

This educational text introduces tensor algebra and analysis in Cartesian and curvilinear coordinates, focusing on their applications in continuum mechanics and differential geometry for graduate students.

Contribution

It provides a focused, application-oriented introduction to tensor calculus tailored for continuum mechanics, emphasizing coordinate transformations and differential geometry.

Findings

Clear presentation of tensor algebra in Cartesian coordinates

Extension to curvilinear coordinates and tensor components

Application to differential geometry of curves and surfaces

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.