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2016 | Buch

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Tensor Calculus for Engineers and Physicists

verfasst von: Emil de Souza Sánchez Filho

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces.

The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step.

Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors.

Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Review of Fundamental Topics About Tensors

Abstract

This chapter presents a brief review of the fundamental concepts required for the consistent development of the later chapters. Various subjects are admitted as being previously known, which allows avoiding demonstrations that overload the text. It is assumed that the reader has full knowledge of Differential and Integral Calculus, Vectorial Calculus, Linear Algebra, and the fundamental concepts about tensors and dominium of the tensorial formalism. However, are presented succinctly the essential topics for understanding the themes that are developed in this book.

Emil de Souza Sánchez Filho

Chapter 2. Covariant, Absolute, and Contravariant Derivatives

Abstract

The curve represented by a function ϕ(x ⁱ) in a closed interval is continuous if this function is continuous in this interval. If the curve is parameterized, i.e., ϕ[x ⁱ(t)] being \( t\in \left[a,b\right] \), then it will be continuous if x ⁱ(t) are continuous functions in this interval, and it will be smooth if it has continuous and non-null derivatives for a value of \( t\in \left[a,b\right] \). The smooth curves do not intersect, i.e., the conditions \( {x}^i(a)={x}^i(b) \) will only be satisfied if \( a=b \). This condition defines a curve that can be divided into differential elements, forming curve arcs. For the case in which the initial and final points coincide, expressed by condition \( a=b \), the curve is closed. The various differential elements obtained on the curve allow calculating its line integral.

Emil de Souza Sánchez Filho

Chapter 3. Integral Theorems

Abstract

The integral theorems and the concepts presented in this chapter are treated in Differential and Integral Calculus of multiple variables.

Emil de Souza Sánchez Filho

Chapter 4. Differential Operators

Abstract

The study of the scalar, vectorial, and tensorial fields is strictly related with the differential operators which are applied to the analytic functions that represent these fields.

Emil de Souza Sánchez Filho

Chapter 5. Riemann Spaces

Abstract

The space provided with metric is called Riemann space, for which the tensorial formalism is based on the study in its first fundamental form, being complemented by the definition of curvature and by the concept of geodesics, which allows expanding the basic conceptions of the Euclidian geometry for this type of space with N dimensions.

Emil de Souza Sánchez Filho

Chapter 6. Geodesics and Parallelism of Vectors

Abstract

The shortest distance between two points located on a surface of the Riemann space E _N is related to a curve of stationary value, which equation is obtained by means of the variational calculus. This curve is called geodesic. The checking of the existence of this type of curve is carried out from the basic concepts of the elementary geometry.

Emil de Souza Sánchez Filho

Backmatter

Titel: Tensor Calculus for Engineers and Physicists
verfasst von: Emil de Souza Sánchez Filho
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-31520-1
Print ISBN: 978-3-319-31519-5
DOI: https://doi.org/10.1007/978-3-319-31520-1

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