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E-raamat: Mathematical Methods for Physics and Engineering [Taylor & Francis e-raamat]

(KTH Royal Institute of Technology, Stockholm, Sweden)
  • Formaat: 736 pages, 247 Illustrations, black and white
  • Ilmumisaeg: 18-Dec-2017
  • Kirjastus: CRC Press
  • ISBN-13: 9781315165097
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  • Taylor & Francis e-raamat
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Mathematical Methods for Physics and EngineeringSuurem pilt
  • Formaat: 736 pages, 247 Illustrations, black and white
  • Ilmumisaeg: 18-Dec-2017
  • Kirjastus: CRC Press
  • ISBN-13: 9781315165097
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781315165097

Suitable for advanced undergraduate and graduate students, this new textbook contains an introduction to the mathematical concepts used in physics and engineering. The entire book is unique in that it draws upon applications from physics, rather than mathematical examples, to ensure students are fully equipped with the tools they need. This approach prepares the reader for advanced topics, such as quantum mechanics and general relativity, while offering examples, problems, and insights into classical physics. The book is also distinctive in the coverage it devotes to modelling, and to oft-neglected topics such as Green's functions.

Chapter 1 Scalars and Vectors 1(68)
1.1 Vectors And Arithmetics
1(2)
1.2 Rotations And Basis Changes
3(3)
1.3 Index Notation
6(4)
1.3.1 The Kronecker delta and the permutation symbol
6(1)
1.3.2 Vector algebra using index notation
7(3)
1.4 Fields
10(14)
1.4.1 Locality
12(1)
1.4.2 Field integrals
13(4)
1.4.2.1 Volume integrals
13(1)
1.4.2.2 Surface integrals
14(2)
1.4.2.3 Line integrals
16(1)
1.4.3 Differential operators and fields
17(7)
1.4.3.1 The gradient
18(1)
1.4.3.2 The divergence
19(1)
1.4.3.3 The curl
20(2)
1.4.3.4 The directional derivative
22(1)
1.4.3.5 Second order operators
22(1)
1.4.3.6 Coordinate independence
23(1)
1.5 Integral Theorems
24(8)
1.5.1 Line integral of a gradient
24(1)
1.5.2 The divergence theorem
25(4)
1.5.3 Green's formula
29(1)
1.5.4 The curl theorem
29(2)
1.5.5 General integral theorems
31(1)
1.6 Non-Cartesian Coordinate Systems
32(17)
1.6.1 General theory
32(6)
1.6.1.1 Tangent vector basis
34(1)
1.6.1.2 Dual basis
35(3)
1.6.2 Orthogonal coordinates
38(4)
1.6.2.1 Integration in orthogonal coordinates
39(1)
1.6.2.2 Differentiation in orthogonal coordinates
40(2)
1.6.3 Polar and cylinder coordinates
42(2)
1.6.4 Spherical coordinates
44(5)
1.7 Potentials
49(10)
1.7.1 Scalar potentials
49(4)
1.7.2 Vector potentials
53(4)
1.7.3 Scalar and vector potentials
57(2)
1.8 Problems
59(10)
Chapter 2 Tensors 69(58)
2.1 Outer Products And Tensor Bases
71(3)
2.1.1 General coordinate bases
72(2)
2.2 Tensor Algebra
74(5)
2.2.1 Tensors and symmetries
75(3)
2.2.2 The quotient law
78(1)
2.3 Tensor Fields And Derivatives
79(17)
2.3.1 The metric tensor
80(3)
2.3.1.1 Distances and the metric tensor
81(1)
2.3.1.2 Lowering and raising indices
82(1)
2.3.2 Derivatives of tensor fields
83(6)
2.3.2.1 The covariant derivative
85(2)
2.3.2.2 Divergence
87(1)
2.3.2.3 Generalised curl
88(1)
2.3.3 Tensor densities
89(4)
2.3.4 The generalised Kronecker delta
93(2)
2.3.5 Orthogonal coordinates
95(1)
2.4 Tensors In Cartesian Coordinates
96(2)
2.5 Tensor Integrals
98(6)
2.5.1 Integration of tensors in Cartesian coordinates
98(4)
2.5.1.1 Volume integration
98(1)
2.5.1.2 Surface integrals
99(1)
2.5.1.3 Line integrals
100(1)
2.5.1.4 Integral theorems
101(1)
2.5.2 The volume element and general coordinates
102(2)
2.6 Tensor Examples
104(15)
2.6.1 Solid mechanics
105(5)
2.6.1.1 The stress tensor
105(2)
2.6.1.2 The strain tensor
107(2)
2.6.1.3 The stiffness and compliance tensors
109(1)
2.6.2 Electromagnetism
110(5)
2.6.2.1 The magnetic field tensor
111(1)
2.6.2.2 The Maxwell stress tensor
111(2)
2.6.2.3 The conductivity and resistivity tensors
113(2)
2.6.3 Classical mechanics
115(19)
2.6.3.1 The moment of inertia tensor
115(2)
2.6.3.2 The generalised inertia tensor
117(2)
2.7 Problems
119(8)
Chapter 3 Partial Differential Equations and Modelling 127(68)
3.1 A Quick Note On Notation
127(1)
3.2 Intensive And Extensive Properties
128(2)
3.3 The Continuity Equation
130(4)
3.4 The Diffusion And Heat Equations
134(4)
3.4.1 Diffusion and Fick's laws
134(2)
3.4.2 Heat conduction and Fourier's law
136(1)
3.4.3 Additional convection currents
137(1)
3.5 The Wave Equation
138(6)
3.5.1 Transversal waves on a string
139(2)
3.5.1.1 Wave equation as an application of continuity
140(1)
3.5.2 Transversal waves on a membrane
141(2)
3.5.3 Electromagnetic waves
143(1)
3.6 Boundary And Initial Conditions
144(7)
3.6.1 Boundary conditions
145(3)
3.6.1.1 Dirichlet conditions
145(1)
3.6.1.2 Neumann conditions
146(1)
3.6.1.3 Robin boundary conditions
147(1)
3.6.2 Initial conditions
148(2)
3.6.3 Uniqueness
150(1)
3.7 PDES In Space Only
151(3)
3.8 Linearisation
154(3)
3.9 The Cauchy Momentum Equations
157(8)
3.9.1 Inviscid fluids
159(2)
3.9.2 Navier-Stokes equations
161(3)
3.9.3 Incompressible flow
164(1)
3.10 Superposition And Inhomogeneities
165(3)
3.10.1 Removing inhomogeneities from boundaries
166(1)
3.10.2 Using known solutions
167(1)
3.11 Modelling Thin Volumes
168(2)
3.12 Dimensional Analysis
170(9)
3.12.1 Units
172(2)
3.12.2 The Buckingham π theorem
174(3)
3.12.3 Dimensional analysis and modelling
177(1)
3.12.4 Parameters as units
178(1)
3.13 Modelling With Delta Functions
179(5)
3.13.1 Coordinate transformations
180(1)
3.13.2 Lines and surfaces
181(3)
3.14 Problems
184(11)
Chapter 4 Symmetries and Group Theory 195(68)
4.1 What Is A Symmetry?
195(3)
4.2 Groups
198(7)
4.2.1 Conjugacy classes
201(2)
4.2.2 Subgroups
203(1)
4.2.3 Homomorphisms
204(1)
4.3 Discrete Groups
205(7)
4.3.1 The cyclic group
207(1)
4.3.2 The dihedral group
208(3)
4.3.2.1 Dihedral groups and three dimensions
209(2)
4.3.3 The symmetric group and permutations
211(1)
4.4 Lie Groups
212(11)
4.4.1 Rotations
215(1)
4.4.2 Translations
216(1)
4.4.3 Matrix groups
217(6)
4.4.3.1 The orthogonal group
218(3)
4.4.3.2 The unitary group
221(2)
4.5 Representation Theory
223(6)
4.5.1 Tensor products and direct sums
224(3)
4.5.2 Reducible representations
227(2)
4.6 Physical Implications And Examples
229(10)
4.6.1 Reduction of possible form of solutions
229(3)
4.6.2 Important transformations in physics
232(7)
4.6.2.1 Time translations and reversal
232(2)
4.6.2.2 Spatial reflections (parity)
234(3)
4.6.2.3 Galilei transformations
237(2)
4.7 IRREPS And Characters
239(14)
4.7.1 Irreducible representations
239(1)
4.7.2 Schur's lemmas and the orthogonality theorem
240(2)
4.7.3 Characters
242(5)
4.7.3.1 Orthogonality of characters
244(1)
4.7.3.2 Decomposition into irreps
245(2)
4.7.4 Physical insights
247(6)
4.8 Outlook
253(1)
4.9 Problems
254(9)
Chapter 5 Function Spaces 263(72)
5.1 Abstract Vector Spaces
263(7)
5.1.1 Inner products and completeness
265(3)
5.1.1.1 Geometry in inner product spaces
266(1)
5.1.1.2 Convergence of series
267(1)
5.1.2 Function spaces as vector spaces
268(2)
5.1.2.1 Inner products on function spaces
269(1)
5.2 Operators And Eigenvalues
270(6)
5.2.1 Application of operators in finite spaces
270(3)
5.2.2 Operators on inner product spaces
273(3)
5.2.2.1 Differential operators and discretisation
274(2)
5.3 Sturm-Liouville Theory
276(7)
5.3.1 Regular Sturm-Liouville problems
277(4)
5.3.1.1 Sturm-Liouville's theorem
279(2)
5.3.2 Periodic and singular Sturm-Liouville problems
281(2)
5.4 Separation Of Variables
283(5)
5.4.1 Separation and Sturm-Liouville problems
284(4)
5.5 Special Functions
288(28)
5.5.1 Polar coordinates
289(11)
5.5.1.1 Bessel functions
291(7)
5.5.1.2 Modified Bessel functions
298(2)
5.5.2 Spherical coordinates
300(13)
5.5.2.1 Legendre polynomials and associated Legendre functions
301(5)
5.5.2.2 Spherical harmonics
306(3)
5.5.2.3 Spherical Bessel functions
309(4)
5.5.3 Hermite functions
313(3)
5.6 Function Spaces As Representations
316(4)
5.6.1 Reducibility
317(3)
5.7 Distribution Theory
320(4)
5.7.1 Distribution derivatives
322(2)
5.8 Problems
324(11)
Chapter 6 Eigenfunction Expansions 335(66)
6.1 Poisson's Equation And Series
335(10)
6.1.1 Inhomogeneous PDE
335(3)
6.1.2 Inhomogeneous boundary conditions
338(4)
6.1.2.1 Transferring inhomogeneities
341(1)
6.1.3 General inhomogeneities
342(3)
6.1.3.1 Superpositions
342(2)
6.1.3.2 Transferring inhomogeneities
344(1)
6.2 Stationary And Steady State Solutions
345(3)
6.2.1 Removing inhomogeneities
346(2)
6.3 Diffusion And Heat Equations
348(10)
6.3.1 Initial conditions
348(3)
6.3.2 Constant source terms
351(2)
6.3.3 Critical systems
353(5)
6.3.4 Time-dependent sources
358(1)
6.4 Wave Equation
358(8)
6.4.1 Inhomogeneous sources and initial conditions
358(3)
6.4.2 Damped systems
361(1)
6.4.3 Driven systems
362(4)
6.5 Terminating The Series
366(5)
6.5.1 Heat and diffusion equations
369(1)
6.5.2 Wave equation
370(1)
6.6 Infinite Domains
371(6)
6.6.1 Domains with a boundary
374(3)
6.6.1.1 The Fourier sine and cosine transforms
375(1)
6.6.1.2 Hankel transforms
376(1)
6.7 Transform Solutions
377(7)
6.7.1 Mixed series and transforms
381(3)
6.8 Discrete And Continuous Spectra
384(4)
6.9 Problems
388(13)
Chapter 7 Green's Functions 401(68)
7.1 What Are Green's Functions?
401(2)
7.2 Green's Functions In One Dimension
403(12)
7.2.1 Inhomogeneous initial conditions
408(4)
7.2.2 Sturm-Liouville operators and inhomogeneities in the boundary conditions
412(2)
7.2.3 The general structure of Green's function solutions
414(1)
7.3 Poisson's Equation
415(6)
7.3.1 Hadamard's method of descent
418(3)
7.4 Heat And Diffusion
421(2)
7.5 Wave Propagation
423(9)
7.5.1 One-dimensional wave propagation
424(2)
7.5.2 Three-dimensional wave propagation
426(2)
7.5.3 Two-dimensional wave propagation
428(1)
7.5.4 Physics discussion
428(4)
7.6 Problems With A Boundary
432(16)
7.6.1 Inhomogeneous boundary conditions
432(2)
7.6.2 Method of images
434(8)
7.6.2.1 Multiple mirrors
438(4)
7.6.3 Spherical boundaries and Poisson's equation
442(4)
7.6.4 Series expansions
446(2)
7.7 Perturbation Theory
448(8)
7.7.1 Feynman diagrams
452(4)
7.8 Problems
456(13)
Chapter 8 Variational Calculus 469(66)
8.1 Functionals
469(2)
8.2 Functional Optimisation
471(9)
8.2.1 Euler-Lagrange equations
473(3)
8.2.1.1 Natural boundary conditions
474(2)
8.2.2 Higher order derivatives
476(2)
8.2.3 Comparison to finite spaces
478(2)
8.3 Constants Of Motion
480(4)
8.3.1 Integrand independent of the function
481(1)
8.3.2 Integrand independent of the variable
482(2)
8.4 Optimisation With Constraints
484(9)
8.4.1 Lagrange multipliers
486(4)
8.4.1.1 Several constraints
489(1)
8.4.2 Isoperimetric constraints
490(2)
8.4.3 Holonomic constraints
492(1)
8.5 Choice Of Variables
493(3)
8.6 Functionals And Higher-Dimensional Spaces
496(6)
8.6.1 Conservation laws
499(3)
8.7 Basic Variational Principles In Physics
502(7)
8.7.1 Fermat's principle
502(3)
8.7.2 Hamilton's principle
505(7)
8.7.2.1 Constants of motion
507(2)
8.8 Modelling With Variational Calculus
509(3)
8.9 Variational Methods In Eigenvalue Problems
512(9)
8.9.1 The Ritz method
515(2)
8.9.2 The Rayleigh-Ritz method
517(21)
8.9.2.1 Finite element method
519(2)
8.10 Problems
521(14)
Chapter 9 Calculus on Manifolds 535(68)
9.1 Manifolds
535(3)
9.2 Formalisation Of Vectors
538(10)
9.2.1 Tangent vectors
540(3)
9.2.1.1 Vector fields
542(1)
9.2.2 Dual vectors
543(3)
9.2.2.1 Differentials as dual vectors
545(1)
9.2.3 Tensors
546(2)
9.3 Derivative Operations
548(17)
9.3.1 The Lie bracket
548(1)
9.3.2 Affine connections
549(5)
9.3.2.1 Coordinate transformations
552(1)
9.3.2.2 Affine connections and tensor fields
553(1)
9.3.3 Parallel fields
554(3)
9.3.4 Torsion
557(4)
9.3.5 Curvature
561(4)
9.4 Metric Tensor
565(8)
9.4.1 Inner products
566(1)
9.4.2 Length of curves
567(3)
9.4.3 The Levi-Civita connection
570(2)
9.4.4 Curvature revisited
572(1)
9.5 Integration On Manifolds
573(14)
9.5.1 Differential forms
574(3)
9.5.1.1 The exterior derivative
576(1)
9.5.2 Integration of differential forms
577(3)
9.5.3 Stokes' theorem
580(5)
9.5.4 The continuity equation revisited
585(18)
9.5.4.1 Flux
585(1)
9.5.4.2 Production, concentration, and continuity
586(1)
9.6 Embeddings
587(5)
9.7 Problems
592(11)
Chapter 10 Classical Mechanics and Field Theory 603(88)
10.1 Newtonian Mechanics
603(12)
10.1.1 Motion of a rigid body
604(2)
10.1.2 Dynamics of a rigid body
606(5)
10.1.3 Dynamics in non-inertial frames
611(4)
10.2 Lagrangian Mechanics
615(17)
10.2.1 Configuration space
616(2)
10.2.2 Finite number of degrees of freedom
618(3)
10.2.3 Non-inertial frames in Lagrangian mechanics
621(1)
10.2.4 Noether's theorem
622(3)
10.2.5 Effective potentials
625(7)
10.3 Central Potentials And Planar Motion
632(11)
10.3.1 The two-body problem and Kepler's laws
638(3)
10.3.2 The restricted three-body problem
641(2)
10.4 Hamiltonian Mechanics
643(18)
10.4.1 Phase space
643(3)
10.4.2 The Hamiltonian
646(3)
10.4.3 Poisson brackets
649(3)
10.4.4 Liouville's theorem
652(3)
10.4.5 Canonical transformations
655(3)
10.4.6 Phase space flows and symmetries
658(3)
10.5 Manifolds And Classical Mechanics
661(8)
10.5.1 The Lagrangian formalism revisited
661(3)
10.5.2 The Hamiltonian formalism revisited
664(5)
10.6 Field Theory
669(7)
10.6.1 Noether's theorem revisited
671(3)
10.6.2 Symmetries of the wave equation
674(2)
10.7 Problems
676(15)
Appendix A Reference material 691(14)
A.1 Groups And Character Tables
691(2)
A.1.1 Cyclic groups
691(1)
A.1.1.1 C2
691(1)
A.1.1.2 C3
691(1)
A.1.1.3 C2v
691(1)
A.1.2 Dihedral groups
692(1)
A.1.2.1 D2
692(1)
A.1.2.2 D3
692(1)
A.1.2.3 D3h
692(1)
A.1.3 Symmetric groups
692(1)
A.1.3.1 S2
692(1)
A.1.3.2 S3
693(1)
A.1.3.3 S4
693(1)
A.2 Differential Operators In Orthogonal Coordinates
693(1)
A.2.1 General expressions
693(1)
A.2.2 Cylinder coordinates
694(1)
A.2.3 Spherical coordinates
694(1)
A.3 Special Functions And Their Properties
694(7)
A.3.1 The Gamma function
694(1)
A.3.2 Bessel functions
695(2)
A.3.2.1 Bessel functions
695(1)
A.3.2.2 Modified Bessel functions
695(1)
A.3.2.3 Integral representations
695(1)
A.3.2.4 Asymptotic form
695(1)
A.3.2.5 Relations among Bessel functions
695(1)
A.3.2.6 Expansions
696(1)
A.3.2.7 Orthogonality relations
696(1)
A.3.2.8 Bessel function zeros
696(1)
A.3.3 Spherical Bessel functions
697(1)
A.3.3.1 Spherical Bessel functions
697(1)
A.3.3.2 Relation to Bessel functions
697(1)
A.3.3.3 Explicit expressions
697(1)
A.3.3.4 Rayleigh formulas
697(1)
A.3.3.5 Relations among spherical Bessel functions
698(1)
A.3.3.6 Orthogonality relations
698(1)
A.3.3.7 Spherical Bessel function zeros
698(1)
A.3.4 Legendre functions
698(2)
A.3.4.1 Legendre functions
698(1)
A.3.4.2 Rodrigues' formula
699(1)
A.3.4.3 Relation among Legendre polynomials
699(1)
A.3.4.4 Explicit expressions
699(1)
A.3.4.5 Associated Legendre functions
699(1)
A.3.4.6 Orthogonality relations
699(1)
A.3.5 Spherical harmonics
700(1)
A.3.5.1 Spherical harmonics
700(1)
A.3.5.2 Expression in terms of associated Legendre functions
700(1)
A.3.5.3 Orthogonality relation
700(1)
A.3.5.4 Parity
700(1)
A.3.6 Hermite polynomials
700(1)
A.3.6.1 Hermite polynomials
700(1)
A.3.6.2 Explicit expressions
701(1)
A.3.6.3 Creation operators
701(1)
A.3.6.4 Hermite functions
701(1)
A.3.6.5 Orthogonality relations
701(1)
A.4 Transform Tables
701(4)
A.4.1 Fourier transform
701(1)
A.4.1.1 Transform table
702(1)
A.4.2 Laplace transform
702(3)
A.4.2.1 Transform table
702(3)
Index 705
Mattias Blennow is an associate professor in the Department of Theoretical physics at KTH Royal Institute of Technology, Stockholm, Sweden. His field of research is directed towards weakly interacting particle physics, specializing in theoretical neutrino and dark matter physics with 51 scientific papers published in peer-reviewed journals. He has taught courses at all university levels, including mathematical methods in physics, quantum mechanics, special and general relativity, and quantum field theory.
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