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2017 | Book

General Basis and Bra-Ket Notation Read chapter Read first chapter

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Authors: Hung Nguyen-Schäfer, Jan-Philip Schmidt

Publisher: Springer Berlin Heidelberg

Book Series : Mathematical Engineering

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book comprehensively presents topics, such as Dirac notation, tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. Additionally, two new chapters of Cartan differential forms and Dirac and tensor notations in quantum mechanics are added to this second edition. 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, differential geometry, and differential forms; and to apply them to the physical and engineering world. Many methods and applications are given in CFD, continuum mechanics, electrodynamics in special relativity, cosmology in the Minkowski four-dimensional spacetime, and relativistic and non-relativistic quantum mechanics.

Tensors, differential geometry, differential forms, and Dirac notation are very useful advanced mathematical tools in many fields of modern physics and computational engineering. They are involved in special and general relativity physics, quantum mechanics, cosmology, electrodynamics, computational fluid dynamics (CFD), and continuum mechanics.

The target audience of this all-in-one book primarily comprises graduate students in mathematics, physics, engineering, research scientists, and engineers.

Table of Contents

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 any arbitrarily chosen coordinate system. Tensors describing these characteristics are invariant under coordinate transformations; however, their tensor components heavily depend on the coordinate bases. Therefore, the tensor components change as the coordinate system varies in the considered spaces. Before going into details, 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 gi be a basis at the point Pi(u1,…,uN) and gj be another basis at the other point Pj(u1,…,uN). 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. The N-tuple of coordinates are invariant in any chosen basis; however, its components on the coordinates change as the coordinate system varies. Therefore, the relating components have to be taken into account by the coordinate transformations.
Hung Nguyen-Schäfer, Jan-Philip Schmidt
Chapter 4. Differential Forms
Abstract
Alternative to tensors, differential forms are very useful in differential geometry without considering the coordinates compared to tensors. Differential forms are based on exterior algebra in which the coordinates are not taken into account. The exterior algebra was developed by Élie Cartan (1869–1951) and Henri Poincaré (1854–1912).
Hung Nguyen-Schäfer, Jan-Philip Schmidt
Chapter 5. 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. The Nabla operator can be usually defined in N-dimensional Cartesian coordinates {xi} using Einstein summation convention as
$$ \nabla \equiv {\mathbf{e}}^i\frac{\partial }{\partial {x}^i}\ \mathrm{f}\mathrm{o}\mathrm{r}\ i = 1,2,\dots, N $$
According to Eq. 2.​12, the relation between the bases of Cartesian and general curvilinear coordinates can be written as
$$ {\mathbf{g}}_i=\frac{\partial \mathbf{r}}{\partial {u}^i}=\frac{\partial \mathbf{r}}{\partial {x}^j}\frac{\partial {x}^j}{\partial {u}^i}={\mathbf{e}}_j\frac{\partial {x}^j}{\partial {u}^i} $$
Multiplying Eq. (5.2) by giej, one obtains the basis of Cartesian coordinates expressed in the curvilinear coordinate basis.
$$ {\mathbf{e}}^j={\mathbf{g}}^i\frac{\partial {x}^j}{\partial {u}^i} $$
Using chain rule of coordinate transformation, the Nabla operator in the general curvilinear coordinates {ui} results from Eq. (5.3) [1, 2].
$$ \begin{array}{c}\nabla \equiv {\mathbf{e}}^i\left(\frac{\partial }{\partial {u}^j}\frac{\partial {u}^j}{\partial {x}^i}\right)={\mathbf{g}}^k\frac{\partial {x}^i}{\partial {u}^k}\left(\frac{\partial }{\partial {u}^j}\frac{\partial {u}^j}{\partial {x}^i}\right)\\ {}={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left(\frac{\partial {x}^i}{\partial {u}^k}\frac{\partial {u}^j}{\partial {x}^i}\right)={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left(\frac{\partial {u}^j}{\partial {u}^k}\right)\\ {}={\mathbf{g}}^k\frac{\partial }{\partial {u}^j}\left({\delta}_k^j\right)={\mathbf{g}}^k\frac{\partial }{\partial {u}^k}\end{array} $$
Thus, the Nabla operator can be written in the curvilinear coordinates {ui}using Einstein summation convention.
$$ \nabla \equiv {\mathbf{g}}^i\frac{\partial }{\partial {u}^i}={\mathbf{g}}^i{\nabla}_i\ \mathrm{f}\mathrm{o}\mathrm{r}\ i = 1,2,\dots, N $$
Hung Nguyen-Schäfer, Jan-Philip Schmidt
Chapter 6. Tensors and Bra-Ket Notation in Quantum Mechanics
Abstract
Classical mechanics have been used to study Newtonian mechanics including conventional mechanics, electrodynamics, cosmology, and relativity physics of Einstein. Compared to quantum mechanics, classical mechanics is deterministic, in which the processes are well determined and foreseeable. On the contrary, processes in quantum mechanics are undeterministic, uncertain, nonlocal, and intrinsically random. Therefore, quantum mechanics deals with probability and uncertainty principles to study the atomic and subatomic world. Due to their extremely small sizes of the quantum particles, the capabilities of apparatuses are quite limited to measure the undeterministic and uncertain behaviors of the particles. Therefore, interactions between these very small particles in the quantum world, such as electrons and photons are simulated using mathematical abstractions that will be discussed in the following sections. Mathematics of quantum physics describes nonlocal correlations between the particles with the intrinsic randomness (e.g. Schrödinger’s cat) in the combined system.
Hung Nguyen-Schäfer, Jan-Philip Schmidt
Backmatter
Metadata
Title
Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers
Authors
Hung Nguyen-Schäfer
Jan-Philip Schmidt
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-48497-5
Print ISBN
978-3-662-48495-1
DOI
https://doi.org/10.1007/978-3-662-48497-5

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