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E-raamat: Invitation to Mathematical Physics and Its History

  • Formaat: PDF+DRM
  • Ilmumisaeg: 22-Sep-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030537593
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Invitation to Mathematical Physics and Its HistorySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 22-Sep-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030537593
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This state of the art book takes an applications based approach to teaching mathematics to engineering and applied sciences students. The book lays emphasis on associating mathematical concepts with their physical counterparts, training students of engineering in mathematics to help them learn how things work. The book covers the concepts of number systems, algebra equations and calculus through discussions on mathematics and physics, discussing their intertwined history in a chronological order. The book includes examples, homework problems, and exercises.  This book can be used to teach a first course in engineering mathematics or as a refresher on basic mathematical physics. Besides serving as core textbook, this book will also appeal to undergraduate students with cross-disciplinary interests as a supplementary text or reader.
1 Introduction
1(18)
1.1 Early Science and Mathematics
2(6)
1.1.1 The Pythagorean Theorem
3(2)
1.1.2 What Is Science?
5(1)
1.1.3 What Is Mathematics?
5(2)
1.1.4 Early Physics as Mathematics: Back to Pythagoras
7(1)
1.2 Modern Mathematics
8(11)
1.2.1 Science Meets Mathematics
9(7)
1.2.2 Three Streams from the Pythagorean Theorem
16(3)
2 Stream 1: Number Systems
19(50)
2.1 The Taxonomy of Numbers: P, N, Z, Q, F, I, R, C
20(11)
2.1.1 Numerical Taxonomy
25(1)
2.1.2 Applications of Integers
26(5)
2.2 Problems NS-1
31(2)
2.2.1 Topic of This Homework
31(1)
2.2.2 Plotting Complex Quantities in Octave/MATLAB
31(1)
2.2.3 Prime Numbers, Infinity, and Special Functions in Octave/MATLAB
31(2)
2.3 The Role of Physics in Mathematics
33(6)
2.3.1 The Three Streams of Mathematics
34(1)
2.3.2 Stream 1: Prime Number Theorems
35(2)
2.3.3 Stream 2: Fundamental Theorem of Algebra
37(1)
2.3.4 Stream 3: Fundamental Theorems of Calculus
37(1)
2.3.5 Other Key Mathematical Theorems
38(1)
2.4 Applications of Prime Numbers
39(12)
2.4.1 The Importance of Prime Numbers
39(2)
2.4.2 Two Fundamental Theorems of Primes
41(1)
2.4.3 Greatest Common Divisor (Euclidean Algorithm)
42(4)
2.4.4 Continued Fraction Algorithm
46(5)
2.5 Problems NS-2
51(5)
2.5.1 Topic of This Homework
51(1)
2.5.2 Prime Numbers
51(1)
2.5.3 Greatest Common Divisors
52(1)
2.5.4 Algebraic Generalization of the GCD (Euclidean) Algorithm
53(1)
2.5.5 Continued Fractions
54(1)
2.5.6 Continued Fraction Algorithm (CFA)
55(1)
2.6 Number Theory Applications
56(8)
2.6.1 Pythagorean Triplets (Euclid's Formula)
56(1)
2.6.2 Pell's Equation
57(3)
2.6.3 Fibonacci Sequence
60(2)
2.6.4 Diagonalization of a Matrix (Eigenvalue/Eigenvector Decomposition)
62(1)
2.6.5 Finding the Eigenvalues
63(1)
2.6.6 Finding the Eigenvectors
64(1)
2.7 Problems NS-3
64(5)
2.7.1 Topic of This Homework
64(1)
2.7.2 Pythagorean Triplets
64(1)
2.7.3 Pell's Equation
65(1)
2.7.4 The Fibonacci Sequence
66(2)
2.7.5 CFA as a Matrix Recursion
68(1)
3 Algebraic Equations: Stream 2
69(108)
3.1 Algebra and Geometry as Physics
69(19)
3.1.1 The First Algebra
73(1)
3.1.2 Finding Roots of Polynomials
74(10)
3.1.3 Matrix Formulation of the Polynomial
84(2)
3.1.4 Working with Polynomials in MATLAB/Octave
86(2)
3.2 Eigenanalysis
88(15)
3.2.1 Eigenvalues of a Matrix
88(2)
3.2.2 Cauchy's Theorem and Eigenmodes
90(3)
3.2.3 Taylor Series
93(5)
3.2.4 Analytic Functions
98(2)
3.2.5 Brune Impedances
100(1)
3.2.6 Complex Analytic Functions
101(2)
3.3 Problems AE-1
103(5)
3.3.1 Polynomials and the Fundamental Theorem of Algebra (FTA)
103(1)
3.3.2 Analytic Functions
104(2)
3.3.3 Inverse Analytic Functions and Composition
106(1)
3.3.4 Convolution
106(1)
3.3.5 Newton's Root-Finding Method
107(1)
3.3.6 Riemann Zeta Function ζ(s)
107(1)
3.4 Root Classification by Convolution
108(6)
3.4.1 Convolution of Monomials
109(3)
3.4.2 Residue Expansions of Rational Functions
112(2)
3.5 Introduction to Analytic Geometry
114(14)
3.5.1 Merging the Concepts
115(5)
3.5.2 Generalized Scalar Product
120(1)
3.5.3 Development of Analytic Geometry
121(2)
3.5.4 Applications of Scalar Products
123(2)
3.5.5 Gaussian Elimination
125(3)
3.6 Matrix Algebra: Systems
128(7)
3.6.1 Vectors
128(1)
3.6.2 Vector Products
129(1)
3.6.3 Norms of Vectors
130(2)
3.6.4 Matrices
132(1)
3.6.5 N x M Complex Matrices
132(3)
3.6.6 Inverse of the 2 &time; 2 Matrix
135(1)
3.7 Problems AE-2
135(8)
3.7.1 Topics of This Homework
135(1)
3.7.2 Gaussian Elimination
136(1)
3.7.3 Two Linear Equations
136(1)
3.7.4 Integer Equations: Applications and Solutions
137(1)
3.7.5 Vector Algebra in R3
138(2)
3.7.6 Ohm's Law
140(1)
3.7.7 Nonlinear (Quadratic) to Linear Equations
141(1)
3.7.8 Nonlinear Intersection in Analytic Geometry
141(2)
3.8 Transmission (ABCD) Matrix Composition Method
143(9)
3.8.1 Thevenin Parameters of a Source
144(1)
3.8.2 The Impedance Matrix
145(2)
3.8.3 Network Power Relationships
147(1)
3.8.4 Ohm's Law and Impedance
148(4)
3.9 Signals: Fourier Transforms
152(5)
3.9.1 Properties of the Fourier Transform
154(3)
3.10 Systems: Laplace Transforms
157(10)
3.10.1 Properties of the Laplace Transform
158(4)
3.10.2 System Postulates
162(3)
3.10.3 Probability
165(2)
3.11 Complex Analytic Mappings (Domain-Coloring)
167(6)
3.11.1 The Riemann Sphere
170(2)
3.11.2 Bilinear Transformation
172(1)
3.12 Problems AE-3
173(4)
3.12.1 Topics of This Homework
173(1)
3.12.2 Two-Port Network Analysis
173(1)
3.12.3 Algebra
174(1)
3.12.4 Algebra with Complex Variables
174(1)
3.12.5 Schwarz Inequality
175(1)
3.12.6 Probability
176(1)
4 Stream 3A: Scalar Calculus
177(50)
4.1 The Beginning of Modern Mathematics
178(2)
4.2 Fundamental Theorem of Scalar Calculus
180(4)
4.2.1 Fundamental Theorem of Real Calculus
180(1)
4.2.2 The Fundamental Theorem of Complex Calculus
181(1)
4.2.3 Cauchy-Riemann Conditions
182(2)
4.3 Problems DE-1
184(3)
4.3.1 Topics of This Homework
184(1)
4.3.2 Complex Power Series
184(1)
4.3.3 Cauchy-Riemann Equations
185(1)
4.3.4 Branch Cuts and Riemann Sheets
186(1)
4.3.5 A Cauer Synthesis of any Brune Impedance
187(1)
4.4 Complex Analytic Brune Admittance
187(16)
4.4.1 Generalized Admittance/Impedance
189(1)
4.4.2 Complex Analytic Impedance
190(5)
4.4.3 Multivalued Functions
195(8)
4.5 Three Cauchy Integral Theorems
203(3)
4.5.1 Cauchy's Theorems for Integration in the Complex Plane
203(1)
4.5.2 Cauchy Integral Formula and Residue Theorem
204(2)
4.6 Problems DE-2
206(10)
4.6.1 Topics of This Homework
206(1)
4.6.2 Two Fundamental Theorems of Calculus
206(2)
4.6.3 Cauchy's Theorems CT-1, CT-2, CT-3
208(2)
4.6.4 Integration of Analytic Functions
210(1)
4.6.5 Laplace Transform Applications
211(1)
4.6.6 Computer Exercises with Matlab/Octave
212(2)
4.6.7 Inverse of Riemann ζ(s) Function
214(2)
4.6.8 Quadratic Forms
216(1)
4.7 The Laplace Transform and Its Inverse
216(7)
4.7.1 Case for Negative Time 0 >0) and Causality
218(1)
4.7.2 Case for Zero Time (t = 0)
218(1)
4.7.3 Case for Positive Time (r < 0)
218(2)
4.7.4 Properties of the CT
220(2)
4.7.5 Solving Differential Equations
222(1)
4.8 Problems DE-3
223(4)
4.8.1 Topics of This Homework: Brune Impedance
223(1)
4.8.2 Brune Impedance
223(1)
4.8.3 Transmission Line Analysis
224(3)
5 Stream 3B: Vector Calculus
227(64)
5.1 Properties of Fields and Potentials
227(15)
5.1.1 Scalar and Vector Fields
227(2)
5.1.2 Gradient δ, Divergence δ, Curl δx, and Laplacian δ2
229(5)
5.1.3 Scalar Laplacian Operator in N Dimensions
234(8)
5.2 Partial Differential Equations and Field Evolution
242(10)
5.2.1 The Laplacian δ2
242(4)
5.2.2 Scalar Wave Equation (Acoustics)
246(2)
5.2.3 The Webster Horn Equation (WHEN)
248(2)
5.2.4 Matrix Formulation of the WHEN
250(2)
5.3 Problems VC-1
252(4)
5.3.1 Topics of This Homework
252(1)
5.3.2 Scalar Fields and the δ Operator
252(1)
5.3.3 Vector Fields and the δ Operator
253(1)
5.3.4 Vector and Scalar Field Identities
254(1)
5.3.5 Integral Theorems
254(1)
5.3.6 System Classification
255(1)
5.4 Three Examples of Finite-Length Horns
256(5)
5.4.1 Uniform Horn
257(2)
5.4.2 Conical Horn
259(1)
5.4.3 Exponential Horn
260(1)
5.5 Solution Methods
261(4)
5.5.1 Eigenfunctions e ±(r,t) of the WHEN
263(2)
5.6 Integral Definitions of δ(), δ-(), δ×(), and δ & carried ()
265(11)
5.6.1 Gradient: E= -δφ(x)
265(1)
5.6.2 Divergence: δ D = ρ [ C/m3]
266(2)
5.6.3 Integral Definition of the Curl: δ×H = C
268(1)
5.6.4 Summary
269(1)
5.6.5 Helmholtz's Decomposition Theorem
270(4)
5.6.6 Second-Order Operators
274(2)
5.7 The Unification of Electricity and Magnetism
276(6)
5.7.1 Field Strength E,H
277(1)
5.7.2 Flux D,B
277(1)
5.7.3 Maxwell's Equations
277(3)
5.7.4 Derivation of the Vector Wave Equation
280(2)
5.8 Potential Solutions of Maxwell's Equations
282(3)
5.8.1 Use of Helmholtz's Theorem on Potential Solutions
283(1)
5.8.2 ME for E(x, t)
283(1)
5.8.3 ME for H(x,t)
284(1)
5.8.4 Summary
284(1)
5.9 Problems VC-2
285(5)
5.9.1 Topics of This Homework
285(1)
5.9.2 Partial Differential Equations (PDEs): Wave Equation
285(1)
5.9.3 Helmholtz's Formula
286(2)
5.9.4 Maxwell's Equations
288(1)
5.9.5 Second-Order Differentials
289(1)
5.9.6 Capacitor Analysis
289(1)
5.9.7 Webster Horn Equation
290(1)
5.10 Further Readings
290(1)
Appendix A Notation 291(14)
Appendix B Eigenanalysis 305(8)
Appendix C Laplace Transforms &Pound;t 313(10)
Appendix D Visco-Thermal Losses 323(6)
Appendix E Thermodynamic Systems 329(6)
Appendix F Number Theory Applications 335(4)
Appendix G Eleven Postulates of Systems of Algebraic Networks 339(8)
Appendix H Webster Horn Equation Derivation 347(6)
Appendix I Quantum Mechanics and the WHEN 353(12)
References 365(6)
Index 371
Jont Allen is a Professor in the Department of Electrical and Computer Engineering, University of Illinois. After completing his Ph.D. from the University of Pennsylvania, Philadelphia in 1970, he went to Bell Labs, where he enjoyed a 32 year AT&T Bell Labs career. At AT&T Allen specialized in nonlinear cochlear modeling, auditory and cochlear speech processing, and speech perception. Since joining University of Illinois in 2003, he has taught and worked with his students on the theory and practice of human speech recognition, for both normal and hearing impaired hearing as well as reading disabilities in young children. Prof Allen has more than 20 US patents on hearing aids, signal processing and middle ear measurement diagnostics.  
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