Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

verfasst von: Hung Nguyen-Schäfer, Jan-Philip Schmidt

Verlag: Springer Berlin Heidelberg

Buchreihe : Mathematical Engineering

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

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Über dieses Buch

Tensors and methods of differential geometry are very useful mathematical tools in many fields of modern physics and computational engineering including relativity physics, electrodynamics, computational fluid dynamics (CFD), continuum mechanics, aero and vibroacoustics and cybernetics.

This book comprehensively presents topics, such as bra-ket notation, tensor analysis and elementary differential geometry of a moving surface. Moreover, authors intentionally abstain from giving mathematically rigorous definitions and derivations that are however dealt with as precisely as possible. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors and differential geometry and to use them in the physical and engineering world. The target audience primarily comprises graduate students in physics and engineering, research scientists and practicing engineers.

Inhaltsverzeichnis

Frontmatter

Chapter 1. General Basis and Bra–Ket Notation

Abstract

We begin this chapter by reviewing some mathematical backgrounds dealing with coordinate transformations and general basis vectors in general curvilinear coordinates. Some of these aspects will be informally discussed for the sake of simplicity. Therefore, those readers interested in more in-depth coverage should consult the literature recommended under Further Reading. To simplify notation, we will denote a basis vector simply as basis in the following section.

Hung Nguyen-Schäfer, Jan-Philip Schmidt

Chapter 2. Tensor Analysis

Abstract

Tensors are a powerful mathematical tool that is used in many areas in engineering and physics including general relativity theory, quantum mechanics, statistical thermodynamics, classical mechanics, electrodynamics, solid mechanics, and fluid dynamics. Laws of physics and physical invariants must be independent of an arbitrarily chosen coordinate system. However, the tensor components describing these characteristics heavily depend on the coordinate bases and therefore change as the coordinate system varies in the considered spaces. Before going into detail, we provide less-experienced readers with some examples.

Hung Nguyen-Schäfer, Jan-Philip Schmidt

Chapter 3. Elementary Differential Geometry

Abstract

We consider an N-dimensional Riemannian manifold M, and let g _i be a basis at the point P _i(u ¹, …, u ^N) and g _j be another basis at the other point P _j(u ¹, …,u ^N). Note that each such basis may only exist in a local neighborhood of the respective points and not necessarily for the whole space. For each such point, we may construct an embedded affine tangential manifold.

Hung Nguyen-Schäfer, Jan-Philip Schmidt

Chapter 4. Applications of Tensors and Differential Geometry

Abstract

Nabla operator is a linear map of an arbitrary tensor into an image tensor in N-dimensional curvilinear coordinates.

Hung Nguyen-Schäfer, Jan-Philip Schmidt

Backmatter

Titel: Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers
verfasst von: Hung Nguyen-Schäfer
Jan-Philip Schmidt
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-43444-4
Print ISBN: 978-3-662-43443-7
DOI: https://doi.org/10.1007/978-3-662-43444-4

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