Bernacchi G. Tensors made easy

6th ed. — lulu.com, 2019. — 181 p.This book aims to be an introduction, the simplest and most informal possible, to Tensor Analysis, whose concepts and methods are illustrated. It is not a popular divulging book, but rather a book that wants to do math, in a somewhat casual and laid-back but not superficial way. Tensor Analysis seems to be made to drown the student in a sea of formulas and a swirl of indexes, leaving implicit or taking for granted the underlying concepts; this also happens in many introductory texts. In this sense "Tensors made easy" wants to set itself in counter-trend. The attempt is to accompany the reader from the beginning to a fair level of knowledge of the topic by giving transparency to the line of reasoning, without taking anything for granted, emphasizing the logical links and giving justification of the statements, even at the cost of some redundancy. Tensors are an important tool in various branches of physics, but completely unavoidable for the understanding of General Relativity. Right at GR this book pays special attention and is proposed as a useful introduction. As a general approach, we fully agree on the opportunity of the "geometrical approach" to the Tensor Analysis: it does not really require more effort, but it gives a solidity to the concept of tensor that the old traditional "by component" approach, still present in many texts, cannot give. The comprehension of the text requires the operational knowledge of the differential calculus, up to the Taylor series and to partial derivatives. Some notions of Linear Algebra can help; about matrices, it will be enough to know that they are tables with rows and columns (and that swapping them a great confusion can be created), or little more. We will constantly use the Einstein sum convention for repeated indexes which greatly simplifies the writing of tensor equations.The author's hope for these notes is that they can be useful for those starting to study the topic.Notations and conventions.
Vectors and Covectors.
Tensors.
Change of basis.
Vector-covector “dual switch” tensor.
Tensors in manifolds.
Curved manifolds.
Appendix.
Bibliographic references.