An Invitation to Mathematical Physics and its History Jont B. Allen Copyright c 2016 Jont Allen Help from Steve Levinson, John D’Angelo, Michael Stone University of Illinois at Urbana-Champaign http://jontalle.web.engr.illinois.edu/uploads/298/ July 14, 2017; Version 0.81.8 2 Contents 1 Introduction 13 1.1 EarlyScienceandMathematics . .......... 15 1.1.1 Lecture 1: (Week 1 ) Three Streams from the Pythagorean theorem . 15 1.1.2 PythagoreanTriplets. ....... 16 1.1.3 Whatismathematics? . ..... 17 1.1.4 Earlyphysicsasmathematics . ........ 17 1.1.5 Thebirthofmodernmathematics . ....... 19 1.1.6 TheThreePythagoreanStreams . ....... 20 1.2 Stream1: NumberSystems(9Lectures) . ........... 21 1.2.1 Lecture 2: The Taxonomy of Numbers: P, N, Z, Q, I, R, C ............. 22 1.2.2 Lecture 3: The role of physics in mathematics . ............ 28 1.2.3 Lecture4:Primenumbers . ..... 31 1.2.4 Lecture 5: Early number theory: The Euclidean algorithm............. 34 1.2.5 Lecture 6: Early number theory: Continued fractions . ............... 36 1.2.6 Lecture 7: Pythagorean triplets (Euclid’s formula) . ................ 38 1.2.7 Lecture 8: (Week 3 )Pell’sEquation ......................... 39 1.2.8 Lecture 9: (Week 4 )Fibonaccisequence . 41 1.2.9 Lecture10:ExamI ............................... 42 1.3 Stream 2: Algebraic Equations (11 Lectures) . .............. 42 1.3.1 Lecture 11: (Week 4) Algebra and mathematics driven by physics . 43 1.3.2 Lecture 12: Polynomial root classification (convolution) .............. 50 1.3.3 Lecture 13: (Week 5 )Pole-Residueexpansions . 54 1.3.4 Lecture 14: Introduction to Analytic Geometry . ............. 55 1.3.5 DevelopmentofAnalyticGeometry. ......... 58 1.3.6 Lecture 15: (Week 6 )GaussianElimination . 59 1.3.7 Lecture 16: Transmission (ABCD) matrix composition method .......... 61 1.3.8 Lecture17:RiemannSphere . ...... 63 1.3.9 Lecture 18: (Week 7 ) Complex analytic mappings (Domain-coloring) . 65 1.3.10 Lecture 19: Signals: Fourier and Laplace transforms ................ 66 1.3.11 Lecture 20: Systems: Laplace transforms . ............ 70 1.3.12 Lecture 21: (Week 8 ) Network (System) Postulates . 72 1.3.13 Lecture22:ExamII ............................. ..... 74 1.4 Stream 3a: Scalar Calculus (12 Lectures) . ............. 74 1.4.1 Lecture 23: (Week 8 ) Fundamental theorems of calculus . 76 1.4.2 Lecture 24: (Week 9 ) Cauchy-Riemann conditions . 77 1.4.3 Lecture 25: Complex Analytic functions and Impedance .............. 78 1.4.4 Lecture 26: Branch cuts, Riemann Sheets . .......... 81 1.4.5 Lecture 27: (Week 10 ) Three complex integration theorems I . 84 1.4.6 Lecture 28: Three complex integration theorems II . .............. 85 1.4.7 Lecture 29: Inverse Laplace transform (Cauchy residuetheorem) . 85 1.4.8 Lecture 30: (Week 11 ) Inverse Laplace transform and the Cauchy Residue Theorem 86 3 4 CONTENTS 1.4.9 Lecture31: PropertiesoftheLT . ........ 87 1.4.10 Lecture32: BruneImpedance . ....... 88 1.4.11 Lecture 33: (Week 12 ) The Riemann Zeta function ζ(s).............. 91 1.4.12 Lecture34:ExamIII ............................ ..... 95 1.5 Stream 3b: Vector Calculus (7 Lectures) . ............. 95 1.5.1 Lecture 35: Gradient, divergence and Laplacian . .............. 95 1.5.2 Lecture 36: (Week 13 )ScalarWaveEquation . 98 1.5.3 Lecture 37: (Week 13 )Websterhornequation. 100 1.5.4 Lecture 38: Integral definitions of (), ∇ () and ∇ () ..............103 ∇ · × 1.5.5 Lecture 39: Second-order operators: Helmholtz’s decomposition . 107 1.5.6 Lecture 40: (Week 14 ) Maxwell’s Equations: The unification E&M . 110 1.5.7 Lecture 41: Quasi-statics and the Wave equation . ............113 1.5.8 Lecture42:Finaloverview . .......115 2 Number Systems: Stream 1 117 2.1 Week2........................................... 117 2.1.1 Lec 4 Twotheoremsonprimes . 117 2.1.2 RSApublic-privatekeys . 118 2.1.3 Lec 5 Greatestcommondivisor(GCD) . 120 2.1.4 Lec 6 Continued Fraction Expansion (CFA) . 121 2.2 Week3........................................... 123 2.2.1 Lec 7 Pythagorean triplets (PTs) and Euclid’s formula . 123 2.2.2 Lec 8 Pell’sEquation .................................125 2.3 Week4........................................... 126 2.3.1 Lec 9 FibonacciNumbers ............................... 126 2.3.2 Lec 10 ExamI .....................................127 3 Algebraic Equations: Stream 2 129 3.1 Week4........................................... 129 3.1.1 Lec 11 Algebraandgeometryasphysics . 129 3.2 Week5........................................... 131 3.2.1 Lec 12 The physics behind complex analytic expressions: linear vs. nonlinear . 131 3.2.2 Lec 13 Rootclassificationofpolynomials . 134 3.2.3 Lec 14 DevelopmentofAnalyticGeometry . 136 3.3 Week6........................................... 139 3.3.1 Lec 15 Gaussian Elimination of linear equations . 139 3.3.2 Lec 16 Matrixcomposition: BilinearandABCD . 141 3.3.3 Lec 17 Introduction to the Branch cut and Riemann sheets . 143 3.4 Week7........................................... 143 3.4.1 Lec 18 Complex analytic mappings (domain coloring) . 143 3.4.2 Lec 19 Signals and Systems: Fourier vs. Laplace Transforms . 144 3.4.3 Lec 20 Role of Causality and the Laplace Transform . 144 3.5 Week8........................................... 145 3.5.1 Lec 21 The 9 postulates of System of algebraic Networks . 145 3.5.2 Lec 22 ExamII(Evening) ..............................150 4 Scalar Calculus: Stream 3a 151 4.0.1 Lec 32 Brune impedance: Working with experimental data . 151 CONTENTS 5 5 Vector Calculus: Stream 3b 153 5.1 Week12.......................................... 153 5.1.1 Lec 35 Scalarvs.Vectorwaveequations . 153 5.2 Week13.......................................... 153 5.2.1 Lec 36 TheWebsterHornequation. 153 5.2.2 Lec 37 Partial differential equations of physics . 160 5.2.3 Lec 38 Gradient, divergence and curl vector operators . 160 5.3 ThanksgivingHoliday ............................. ........160 5.4 Week14.......................................... 160 5.4.1 Lec 39 Geometry of Gradient, divergence and curl vector operators .......160 5.4.2 Lec 40 IntroductiontoMaxwell’sEquation . 161 5.4.3 Lec 41 The Fundamental theorem of Vector Calculus . 161 5.5 Week15.......................................... 161 5.5.1 Lec 42: The Quasi-static approximation and applications . 161 5.5.2 Lec 43: Last day of class: Review of Fund Thms of Mathematics . 162 A Notation 165 A.1 Numbersystems ................................... 165 A.1.1 Units ......................................... 165 A.1.2 Symbolsandfunctions. 165 A.1.3 Greekletters .................................. 165 A.1.4 Table of double-bold numbernotation ........................ 166 A.2 Complex vectors and the impedance matrix . ............167 A.2.1 Vectors in R3 ......................................167 A.3 Matrices........................................ 168 A.4 Periodicfunctions ............................... ........168 A.5 Differential equations vs. Polynomials . ..............169 B Linear algebra of 2x2 matrices 171 B.1 Notation........................................ 171 B.1.1 Gaussianeliminationexercises . ..........173 B.2 Inverseofthe2x2matrix . ........174 B.2.1 Derivationoftheinverseofa2x2matrix. ...........174 C Eigenvector analysis 175 D Solution to Pell’s Equation (N=2) 179 D.1 PellequationforN=3 .............................. .......179 E Laplace transforms 181 F Transmission lines 183 F.1 Transferfunctions ............................... ........183 F.2 Thetransmissionmatrix. .........183 G 2D parabolic horn 187 G.1 3DConicalHorn ................................... 188 G.2 ExponentialHorn................................. .......190 6 CONTENTS H Derivation of the Webster Horn Equation 193 H.1 Theinverseproblem ............................... .......194 H.2 WKBmethod....................................... 194 H.3 Rydbergseries ................................... 195 H.4 Laplacian operator in N dimensions .............................195 I Filter classification 197 I.1 Filterproperties ................................ ........197 I.2 Morequestions ................................... 197 I.3 Entropyanalysis ................................. .......198 J Stillwell’s Intersection vs. composition 199 CONTENTS 7 Abstract An understanding of physics requires knowledge of mathematics. The contrary is not true. By defi- nition, pure mathematics contains no physics. Yet historically, mathematics has a rich history filled with physical applications. Mathematics was developed by people with intent of making things work. In my view, as an engineer, I see these creators of early mathematics, as budding engineers. This book 5 is an attempt to tell this story, of the development of mathematical physics, as viewed by an engineer. The book is broken down into three topics, called streams, presented as five chapters: 1) Intro- duction, 2) Number systems, 3) Algebra Equations, 4) Scalar Calculus, and 5) Vector Calculus. The material is delivered as 40 “Lectures” spread out over a semester of 15 weeks, three lectures per week, with a 3 lecture time-out for administrative duties. Problems are provided for each week’s assign- 10 ment. These problems are written out in LATEX, with built in solutions, that may be expressed by un-commenting one line. Once the home-works are turned in, each student is given the solution. The students rated these Assignments as the most important part of the course. There is a built in interplay between these assignments and the lectures. On many occasions I solved the homework in class, as motivation for coming to class. 15 There were four exams,