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Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Part of the Mathematical Engineering Book Series

DOI: 10.1007/978-3-662-48497-5

ISBN: 978-3-662-48495-1

Publisher: Springer Berlin Heidelberg - Springer Journals

Aug 17, 2016

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Chapter 1General Basis and Bra-Ket Notation

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 1 –34

[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...
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Chapter 2Tensor Analysis

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 35 –101

[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...
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Chapter 3Elementary Differential Geometry

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 103 –154

[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...
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Chapter 4Differential Forms

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 155 –180

[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...
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Chapter 5Applications of Tensors and Differential Geometry

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 181 –247

[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∇ ≡ ei∂∂xifori=1,2,…,N\documentclass[12pt]{minimal}...
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Chapter 6Tensors and Bra-Ket Notation in Quantum Mechanics

Nguyen-Schäfer, Hung; Schmidt, Jan-Philip

Pages: 249 –311

[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...

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