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Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 2nd ed. 2017 [Kietas viršelis]

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  • Formatas: Hardback, 376 pages, aukštis x plotis: 235x155 mm, weight: 7155 g, 73 Illustrations, black and white; XVII, 376 p. 73 illus., 1 Hardback
  • Serija: Mathematical Engineering
  • Išleidimo metai: 05-Sep-2016
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3662484951
  • ISBN-13: 9783662484951
Kitos knygos pagal šią temą:
Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 2nd ed. 2017Didesnis paveikslėlis
  • Formatas: Hardback, 376 pages, aukštis x plotis: 235x155 mm, weight: 7155 g, 73 Illustrations, black and white; XVII, 376 p. 73 illus., 1 Hardback
  • Serija: Mathematical Engineering
  • Išleidimo metai: 05-Sep-2016
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3662484951
  • ISBN-13: 9783662484951
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783662484951.html
Raktažodžiai:
This book presents tensors and differential geometry in a comprehensive and approachable manner, providing a bridge from the place where physics and engineering mathematics end, and the place where tensor analysis begins.  

Among the topics examined are tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. The book includes numerous examples with solutions and concrete calculations, which guide readers through these complex topics step by step. Mindful of the practical needs of engineers and physicists, book favors simplicity over a more rigorous, formal approach. The book shows readers how to work with tensors and differential geometry and how to apply them to modeling the physical and engineering world.

The authors provide chapter-length treatment of topics at the intersection of advanced mathematics, and physics and engineering:  General Basis and Bra-Ket Notation

Tensor Analysis

Elementary Differential Geometry

Differential Forms

Applications of Tensors and Differential Geometry

Tensors and Bra-Ket Notation in Quantum Mechanics

The text reviews methods and applications in computational fluid dynamics; continuum mechanics; electrodynamics in special relativity; cosmology in the Minkowski four-dimensional space time; and relativistic and non-relativistic quantum mechanics.

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers benefits research scientists and practicing engineers in a variety of fields, who use tensor analysis and differential geometry in the context of applied physics, and electrical and mechanical engineering. It will also interest graduate students in applied physics and engineering.

Recenzijos

The overall impression of the book is positive. Being addressed to physicists and engineers, it succeeds to impart a fairly sound knowledge of differential geometry and its instruments the tensors and show how this theory can be fruitfully applied. (Eleutherius Symeonidis, zbMATH 1369.53002, 2017)

This book is an excellent systematic realization of tensor analysis for engineers and physicists at all levels, from undergraduate students to experts. (Hamid R. Noori, Computing Reviews, September, 2015)

1 General Basis and Bra-Ket Notation
1(34)
1.1 Introduction to General Basis and Tensor Types
1(1)
1.2 General Basis in Curvilinear Coordinates
2(9)
1.2.1 Orthogonal Cylindrical Coordinates
5(3)
1.2.2 Orthogonal Spherical Coordinates
8(3)
1.3 Eigenvalue Problem of a Linear Coupled Oscillator
11(4)
1.4 Notation of Bra and Ket
15(1)
1.5 Properties of Kets
15(1)
1.6 Analysis of Bra and Ket
16(15)
1.6.1 Bra and Ket Bases
16(2)
1.6.2 Gram-Schmidt Scheme of Basis Orthonormalization
18(1)
1.6.3 Cauchy-Schwarz and Triangle Inequalities
19(1)
1.6.4 Computing Ket and Bra Components
19(1)
1.6.5 Inner Product of Bra and Ket
20(2)
1.6.6 Outer Product of Bra and Ket
22(1)
1.6.7 Ket and Bra Projection Components on the Bases
23(1)
1.6.8 Linear Transformation of Kets
23(2)
1.6.9 Coordinate Transformations
25(4)
1.6.10 Hermitian Operator
29(2)
1.7 Applying Bra and Ket Analysis to Eigenvalue Problems
31(4)
References
34(1)
2 Tensor Analysis
35(68)
2.1 Introduction to Tensors
35(1)
2.2 Definition of Tensors
36(2)
2.2.1 An Example of a Second-Order Covariant Tensor
38(1)
2.3 Tensor Algebra
38(27)
2.3.1 General Bases in General Curvilinear Coordinates
38(9)
2.3.2 Metric Coefficients in General Curvilinear Coordinates
47(4)
2.3.3 Tensors of Second Order and Higher Orders
51(4)
2.3.4 Tensor and Cross Products of Two Vectors in General Bases
55(2)
2.3.5 Rules of Tensor Calculations
57(8)
2.4 Coordinate Transformations
65(9)
2.4.1 Transformation in the Orthonormal Coordinates
65(3)
2.4.2 Transformation of Curvilinear Coordinates in EN
68(2)
2.4.3 Examples of Coordinate Transformations
70(2)
2.4.4 Transformation of Curvilinear Coordinates in RN
72(2)
2.5 Tensor Calculus in General Curvilinear Coordinates
74(29)
2.5.1 Physical Component of Tensors
74(2)
2.5.2 Derivatives of Covariant Bases
76(2)
2.5.3 Christoffel Symbols of First and Second Kind
78(1)
2.5.4 Prove That the Christoffel Symbols Are Symmetric
79(1)
2.5.5 Examples of Computing the Christoffel Symbols
80(2)
2.5.6 Coordinate Transformations of the Christoffel Symbols
82(2)
2.5.7 Derivatives of Contravariant Bases
84(1)
2.5.8 Derivatives of Covariant Metric Coefficients
85(1)
2.5.9 Covariant Derivatives of Tensors
86(4)
2.5.10 Riemann-Christoffel Tensor
90(4)
2.5.11 Ricci's Lemma
94(1)
2.5.12 Derivative of the Jacobian
95(2)
2.5.13 Ricci Tensor
97(2)
2.5.14 Einstein Tensor
99(2)
References
101(2)
3 Elementary Differential Geometry
103(52)
3.1 Introduction
103(1)
3.2 Arc Length and Surface in Curvilinear Coordinates
103(4)
3.3 Unit Tangent and Normal Vector to Surface
107(1)
3.4 The First Fundamental Form
108(2)
3.5 The Second Fundamental Form
110(3)
3.6 Gaussian and Mean Curvatures
113(4)
3.7 Riemann Curvature
117(3)
3.8 Gauss-Bonnet Theorem
120(2)
3.9 Gauss Derivative Equations
122(1)
3.10 Weingarten's Equations
123(1)
3.11 Gauss-Codazzi Equations
124(2)
3.12 Lie Derivatives
126(14)
3.12.1 Vector Fields in Riemannian Manifold
127(1)
3.12.2 Lie Bracket
128(1)
3.12.3 Lie Dragging
129(1)
3.12.4 Lie Derivatives
130(8)
3.12.5 Torsion and Curvature in a Distorted and Curved Manifold
138(1)
3.12.6 Killing Vector Fields
138(2)
3.13 Invariant Time Derivatives on Moving Surfaces
140(6)
3.13.1 Invariant Time Derivative of an Invariant Field
141(3)
3.13.2 Invariant Time Derivative of Tensors
144(2)
3.14 Tangent, Cotangent Bundles and Manifolds
146(2)
3.15 Levi-Civita Connection on Manifolds
148(7)
References
153(2)
4 Differential Forms
155(26)
4.1 Introduction
155(2)
4.2 Definitions of Spaces on the Manifold
157(1)
4.3 Differential k-Forms
157(4)
4.4 The Notation ω · X
161(2)
4.5 Exterior Derivatives
163(3)
4.6 Interior Product
166(2)
4.7 Pullback Operator of Differential Forms
168(2)
4.8 Pushforward Operator of Differential Forms
170(1)
4.9 The Hodge Star Operator
171(10)
4.9.1 Star Operator in Vector Calculus and Differential Forms
173(2)
4.9.2 Star Operator and Inner Product
175(2)
4.9.3 Star Operator in the Minkowski Spacetime
177(3)
References
180(1)
5 Applications of Tensors and Differential Geometry
181(68)
5.1 Nabla Operator in Curvilinear Coordinates
181(1)
5.2 Gradient, Divergence, and Curl
182(8)
5.2.1 Gradient of an Invariant
182(1)
5.2.2 Gradient of a Vector
183(1)
5.2.3 Divergence of a Vector
184(2)
5.2.4 Divergence of a Second-Order Tensor
186(2)
5.2.5 Curl of a Covariant Vector
188(2)
5.3 Laplacian Operator
190(2)
5.3.1 Laplacian of an Invariant
190(1)
5.3.2 Laplacian of a Contravariant Vector
191(1)
5.4 Applying Nabla Operators in Spherical Coordinates
192(5)
5.4.1 Gradient of an Invariant
193(2)
5.4.2 Divergence of a Vector
195(1)
5.4.3 Curl of a Vector
196(1)
5.5 The Divergence Theorem
197(7)
5.5.1 Gauss and Stokes Theorems
197(2)
5.5.2 Green's Identities
199(1)
5.5.3 First Green's Identity
199(1)
5.5.4 Second Green's Identity
200(1)
5.5.5 Differentials of Area and Volume
201(1)
5.5.6 Calculating the Differential of Area
201(1)
5.5.7 Calculating the Differential of Volume
202(2)
5.6 Governing Equations of Computational Fluid Dynamics
204(9)
5.6.1 Continuity Equation
204(2)
5.6.2 Navier-Stokes Equations
206(4)
5.6.3 Energy (Rothalpy) Equation
210(3)
5.7 Basic Equations of Continuum Mechanics
213(11)
5.7.1 Cauchy's Law of Motion
213(5)
5.7.2 Principal Stresses of Cauchy's Stress Tensor
218(2)
5.7.3 Cauchy's Strain Tensor
220(3)
5.7.4 Constitutive Equations of Elasticity Laws
223(1)
5.8 Maxwell's Equations of Electrodynamics
224(15)
5.8.1 Maxwell's Equations in Curvilinear Coordinate Systems
225(2)
5.8.2 Maxwell's Equations in the Four-Dimensional Spacetime
227(5)
5.8.3 The Maxwell's Stress Tensor
232(5)
5.8.4 The Poynting's Theorem
237(2)
5.9 Einstein Field Equations
239(2)
5.10 Schwarzschild's Solution of the Einstein Field Equations
241(2)
5.11 Schwarzschild Black Hole
243(6)
References
246(3)
6 Tensors and Bra-Ket Notation in Quantum Mechanics
249(64)
6.1 Introduction
249(1)
6.2 Quantum Entanglement and Nonlocality
250(2)
6.3 Alternative Interpretation of Quantum Entanglement
252(2)
6.4 The Hilbert Space
254(1)
6.5 State Vectors and Basis Kets
255(9)
6.6 The Pauli Matrices
264(1)
6.7 Combined State Vectors
265(5)
6.8 Expectation Value of an Observable
270(3)
6.9 Probability Density Operator
273(6)
6.9.1 Density Operator of a Pure Subsystem
273(2)
6.9.2 Density Operator of an Entangled Composite System
275(4)
6.10 Heisenberg's Uncertainty Principle
279(7)
6.11 The Wave-Particle Duality
286(11)
6.11.1 De Broglie Wavelength Formula
287(2)
6.11.2 The Compton Effect
289(4)
6.11.3 Double-Slit Experiments with Electrons
293(4)
6.12 The Schrodinger Equation
297(9)
6.12.1 Time Evolution in Quantum Mechanics
298(2)
6.12.2 The Schrodinger and Heisenberg Pictures
300(2)
6.12.3 Time-Dependent Schrodinger Equation (TDSE)
302(4)
6.12.4 Discussions of the Schrodinger Wave Functions
306(1)
6.13 The Klein-Gordon Equation
306(2)
6.14 The Dirac Equation
308(5)
References
311(2)
Appendix A Relations Between Covariant and Contravariant Bases 313(6)
Appendix B Physical Components of Tensors 319(4)
Appendix C Nabla Operators 323(6)
Appendix D Essential Tensors 329(6)
Appendix E Euclidean and Riemannian Manifolds 335(24)
Appendix F Probability Function for the Quantum Interference 359(2)
Appendix G Lorentz and Minkowski Transformations in Spacetime 361(4)
Appendix H The Law of Large Numbers in Statistical Mechanics 365(4)
Mathematical Symbols in This Book 369(2)
Further Reading 371(2)
Index 373
Dr. Hung Nguyen-Schäfer is a senior technical manager in development of electric machines for hybrid and electric vehicles at EM-motive GmbH, a joint company of Daimler and Bosch in Germany. He received B.Sc. and M.Sc. in mechanical engineering with nonlinear vibrations in fluid mechanics from the University of Karlsruhe (KIT), Germany in 1985; and a Ph.D. degree in nonlinear thermo- and fluid dynamics from the same university in 1989. He joined Bosch Company and worked as a technical manager on many development projects. Between 2007 and 2013, he was in charge of rotordynamics, bearings and design platforms of automotive turbochargers at Bosch Mahle Turbo Systems in Stuttgart.

He is also the author of two professional engineering books: Rotordynamics of Automotive Turbochargers, Springer (2012) and Aero and Vibroacoustics of Automotive Turbochargers, Springer (2013). 

Rotordynamics of Automotive Turbochargers, Springer (2012) and Aero and Vibroacoustics of Automotive Turbochargers, Springer (2013).

Dr. Jan-Philip Schmidt is a mathematician. He studied mathematics, physics, and economics at the University of Heidelberg, Germany. He received a Ph.D. degree in mathematics from the University of Heidelberg in 2012. His doctoral thesis was funded by a research fellowship from the Heidelberg Academy of Sciences, in collaboration with the Interdisciplinary Center for Scientific Computing (IWR) at the University of Heidelberg. His academic working experience comprises several research visits in France and Israel, as well as project works at the Max-Planck-Institute for Mathematics in the Sciences (MPIMIS) in Leipzig, and at the Max-Planck-Institute for Molecular Genetics (MPIMG) in Berlin. He also worked as a research associate in the AVACS program at Saarland University, Cluster of Excellence (MMCI).