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2017 | Buch

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Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds

verfasst von: Uwe Mühlich

Verlag: Springer International Publishing

Buchreihe : Solid Mechanics and Its Applications

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

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Über dieses Buch

This book presents the fundamentals of modern tensor calculus for students in engineering and applied physics, emphasizing those aspects that are crucial for applying tensor calculus safely in Euclidian space and for grasping the very essence of the smooth manifold concept.

After introducing the subject, it provides a brief exposition on point set topology to familiarize readers with the subject, especially with those topics required in later chapters.

It then describes the finite dimensional real vector space and its dual, focusing on the usefulness of the latter for encoding duality concepts in physics. Moreover, it introduces tensors as objects that encode linear mappings and discusses affine and Euclidean spaces. Tensor analysis is explored first in Euclidean space, starting from a generalization of the concept of differentiability and proceeding towards concepts such as directional derivative, covariant derivative and integration based on differential forms.

The final chapter addresses the role of smooth manifolds in modeling spaces other than Euclidean space, particularly the concepts of smooth atlas and tangent space, which are crucial to understanding the topic. Two of the most important concepts, namely the tangent bundle and the Lie derivative, are subsequently worked out.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The introduction aims to remind the reader that engineering mechanics is derived from classical mechanics, which is a discipline of general physics. Therefore, engineering mechanics also relies on a proper model for space, and the relations between space and geometry are discussed briefly. The idea of expressing geometrical concepts by means of linear algebra is sketched together with the concept of vectors as geometrical objects. Although this book provides only the very first steps of the manifold concept, this chapter intends to make its importance for modern continuum mechanics clear by raising a number of questions which cannot be answered by the conventional approach. Furthermore, aspects regarding mathematical notation used in subsequent chapters are discussed briefly.

Uwe Mühlich

Chapter 2. Notes on Point Set Topology

Abstract

The chapter provides a brief exposition of point set topology. In particular, it aims to make readers from the engineering community feel comfortable with the subject, especially with those topics required in latter chapters. The implicit appearance of topological concepts in the context of continuum mechanics is sketched first. Afterwards, topological concepts like interior and boundary of sets, continuity of mappings, etc., are discussed within metric spaces before the introduction of the concept of topological space.

Uwe Mühlich

Chapter 3. The Finite-Dimensional Real Vector Space

Abstract

The chapter introduces the notion of the finite-dimensional real vector space together with fundamental concepts like linear independence, vector space basis, and vector space dimension. The discussion of linear mappings between vector spaces prepares the ground for introducing the dual space and its basis. Finally, inner product space and reciprocal basis are contrasted with dual space and the corresponding dual basis.

Uwe Mühlich

Chapter 4. Tensor Algebra

Abstract

Dyadic product and tensors are introduced in the context of bilinear forms before extending this scheme to arbitrary but finite dimensions. Afterwards, tensor product spaces are defined. The exterior product is motivated within this chapter by the aim to generalize the notion of volume for arbitrary dimensions and to overcome the limitations implied by the cross product of conventional vector calculus. Within this context, symmetric and skew-symmetric tensors, as well as a generalized version of the Kronecker symbol, are discussed. Furthermore, basic aspects of the so-called star-operator are examined. The latter relates spaces of alternating tensors of equal dimension based on the existence of an inner product.

Uwe Mühlich

Chapter 5. Affine Space and Euclidean Space

Abstract

Affine and euclidean space are discussed primarily in view of their use as models for physical space. Affine mappings and coordinate charts generated by them are examined. Furthermore, topological aspects are addressed.

Uwe Mühlich

Chapter 6. Tensor Analysis in Euclidean Space

Abstract

After briefly reviewing differentiability in \(\mathbb {R}\), a generalization of differentiability based on the directional derivative in \(\mathbb {R}^N\) is established. Gradients of scalar and vector fields are first discussed in \(\mathbb {R}^N\) before adapting these concepts for general euclidean spaces in combination with global charts. Afterwards, nonlinear chart relations, also known as curvilinear coordinates, are examined, and the concept of tangent space at a point is introduced. In this context, the covariant derivative is derived, insinuating its character as special case of the covariant derivative for smooth manifolds. Aspects of integration based on differential forms are discussed together with the exterior derivative and Stoke’s theorem in \(\mathbb {R}^N\).

Uwe Mühlich

Chapter 7. A Primer on Smooth Manifolds

Abstract

Within this chapter, a generalization of the tangent space concept, as well as the notion of a smooth atlas, is introduced in the context of analysis on curves in \(\mathbb {R}^2\) and analysis on surfaces in \(\mathbb {R}^3\). Implications of a complete avoidance of an embedding space, the last step in the transition to smooth manifolds, are discussed, focusing on abstraction level and topology. Furthermore, the notion of the tangent bundle is introduced in the context of vector fields defined on smooth manifolds. After introducing the Lie derivative, a guideline for studying the subject further is provided. Eventually, a selection of further literature is proposed.

Uwe Mühlich

Backmatter

Titel: Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds
verfasst von: Uwe Mühlich
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-56264-3
Print ISBN: 978-3-319-56263-6
DOI: https://doi.org/10.1007/978-3-319-56264-3

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