Название: Tensor Algebra and Analysis for Engineers: With Applications to Differential Geometry of Curves and Surfaces
Автор: Paolo Vannucci
Издательство: World Scientific Publishing
Год: 2023
Страниц: 230
Язык: английский
Формат: pdf (true)
Размер: 11.4 MB

In modern theoretical and applied mechanics, tensors and differential geometry are two almost essential tools. Unfortunately, in university courses for engineering and mechanics students, these topics are often poorly treated or even completely ignored. At the same time, many existing, very complete texts on tensors or differential geometry are so advanced and written in abstract language that discourage young readers looking for an introduction to these topics specifically oriented to engineering applications.This textbook, mainly addressed to graduate students and young researchers in mechanics, is an attempt to fill the gap. Its aim is to introduce the reader to the modern mathematical tools and language of tensors, with special applications to the differential geometry of curves and surfaces in the Euclidean space. The exposition of the matter is sober, directly oriented to problems that are ordinarily found in mechanics and engineering. Also, the language and symbols are tailored to those usually employed in modern texts of continuum mechanics.

Though not exhaustive, as any primer textbook, this volume constitutes a coherent, self-contained introduction to the mathematical tools and results necessary in modern continuum mechanics, concerning vectors, 2nd- and 4th-rank tensors, curves, fields, curvilinear coordinates, and surfaces in the Euclidean space. More than 100 exercises are proposed to the reader, many of them complete the theoretical part through additional results and proofs. To accompany the reader in learning, all the exercises are entirely developed and solved at the end of the book.

In Chapter 1, vectors and points are introduced and also, with a small anticipation of some results of the second chapter, applied vectors are visited.

Chapter 2 is completely devoted to the algebra of second-rank tensors and the succeeding Chapter 3 to that of fourth-rank tensors.
Intentionally, these are the only two types of tensors introduced in the book: They are the most important tensors in mechanics, and they allow us to represent deformation, stress, and the constitutive laws. I preferred not to introduce tensors in an absolutely general way but to go directly to the most important tensors for applications in mechanics; for the same reason, the algebra of other tensors, namely of third-rank tensors, is not presented in this primer text.

The analysis of tensors is done using first-differential geometry of curves, in Chapter 4, for differentiation and integration with respect to only one variable, then introducing the differential operators for fields and deformations, in Chapter 5.

Then, a generalization of second-rank tensor algebra and analysis in the sense of the use of curvilinear coordinates is presented in Chapter 6, where the notion of metric tensor, co- and contravariant components, and Christoffel’s symbols are introduced.

Finally, Chapter 7 is entirely devoted to an introduction to the differential geometry of surfaces.




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