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Vector Calculus (7 Lectures)

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Vector Calculus (7 Lectures) 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 January 2, 2017 2 Contents 1 Introduction 15 1.1 EarlyScienceandMathematics . .......... 17 1.1.1 Lecture 1: (Week 1 ) Three Streams from the Pythagorean theorem . 17 1.1.2 PythagoreanTriplets. ....... 18 1.1.3 Whatismathematics? . ..... 19 1.1.4 EarlyPhysicsasMathematics. ........ 19 1.1.5 Thebirthofmodernmathematics . ....... 21 1.1.6 TheThreePythagoreanStreams . ....... 22 1.2 Stream1: NumberSystems(9Lectures) . ........... 23 1.2.1 Lecture 2: The Taxonomy of Numbers: P, N, Z, Q, I, R, C ............. 24 1.2.2 Lecture 3: The role of physics in mathematics . ............ 28 1.2.3 Lecture4:Primenumbers . ..... 32 1.2.4 Lecture 5: Early number theory: The Euclidean algorithm............. 35 1.2.5 Lecture 6: Early number theory: Continued fractions . ............... 36 1.2.6 Labor day: Week 3 ................................... 39 1.2.7 Lecture 7: Pythagorean triplets (Euclid’s formula) . ................ 39 1.2.8 Lecture 8: (Week 3 )Pell’sEquation ......................... 39 1.2.9 Lecture 9: (Week 4 )Fibonaccisequence . 42 1.2.10 Lecture10:ExamI .............................. 43 1.3 Stream 2: Algebraic Equations (11 Lectures) . .............. 43 1.3.1 Lecture 11: (Week 4) Algebra and mathematics driven by physics . 43 1.3.2 Lecture 12: (Week 5 ) Examples of nonlinear algebra in Physics . 47 1.3.3 Lecture 13: Polynomial root classification (convolution) .............. 48 1.3.4 Lecture 14: Introduction to Analytic Geometry . ............. 50 1.3.5 Lecture 15: (Week 6 )GaussianElimination . 53 1.3.6 Lecture 16: Transmission (ABCD) matrix composition method .......... 54 1.3.7 Lecture17:RiemannSphere . ...... 56 1.3.8 Lecture 18: (Week 7 ) Complex analytic mappings (Domain coloring) . 58 1.3.9 Lecture 19: Signals: Fourier transforms . ............ 58 1.3.10 Lecture 20: Systems: Laplace transforms . ............ 63 1.3.11 Lecture 21: (Week 8 ) Network (System) Postulates . 64 1.3.12 Lecture22:ExamII ............................. ..... 65 1.4 Stream 3a: Scalar Calculus (12 Lectures) . ............. 65 1.4.1 Lecture 23: (Week 8 ) Integration in the complex plane . 66 1.4.2 Lecture 24: (Week 9 ) Cauchy-Riemann conditions . 68 1.4.3 Lecture 25: Complex Analytic functions and Impedance .............. 69 1.4.4 Lecture 26: Branch cuts, Riemann Sheets . .......... 71 1.4.5 Lecture 27: (Week 10 ) Three complex integration theorems I . 74 1.4.6 Lecture 28: Three complex integration theorems II . .............. 75 1.4.7 Lecture 29: Inverse Laplace transform (Cauchy residuetheorem) . 75 1.4.8 Lecture 30: (Week 11 ) Inverse Laplace transform and the Cauchy Residue Theorem 76 3 4 CONTENTS 1.4.9 Lecture31: PropertiesoftheLT . ........ 77 1.4.10 Lecture32: BruneImpedance . ....... 78 1.4.11 Lecture 33: (Week 12 ) The Riemann Zeta function ζ(s).............. 82 1.4.12 Lecture34:ExamIII ............................ ..... 85 1.5 Stream 3b: Vector Calculus (7 Lectures) . ............. 85 1.5.1 Lecture 35: (Week 13 )ScalarWaveEquation . 85 1.5.2 Lecture 36: (Week 13 )Websterhornequation. 86 1.5.3 Alternate forms of the Webster Horn Equation . ........... 88 1.5.4 General solutions of the Horn Equation . .......... 89 1.5.5 Primitive solutions ̺±(x,t) .............................. 89 1.5.6 Theuniformhorn ................................ 94 1.5.7 Lecture 37: Gradient, divergence and Laplacian; Gauss’Law ........... 95 1.5.8 Lecture 38: Curl, scaler & Vector Laplacian Part; Stoke’sLaw .......... 96 1.5.9 Lecture 39: (Week 14 ) Maxwell’s Equations: The unification E&M . 99 1.5.10 Lecture 40: The fundamental theorem of vector calculus..............100 1.5.11 Lecture 41: Quasi-statics and the Wave equation . .............103 1.5.12 Lecture42: Finaloverview . ........105 2 Number Systems: Stream 1 107 2.1 Week2........................................... 107 2.1.1 Lec 4 Twotheoremsonprimes . 107 2.1.2 Lec 5 Greatestcommondivisor(GCD) . 109 2.1.3 Lec 6 Continued Fraction Expansion (CFA) . 111 2.2 Week3........................................... 112 2.2.1 Lec 7 Pythagorean triplets (PTs) and Euclid’s formula , . 112 2.2.2 Lec 8 Pell’sEquation .................................115 2.3 Week4........................................... 116 2.3.1 Lec 9 FibonacciNumbers ............................... 116 2.3.2 Lec 10 ExamI .....................................117 3 Algebraic Equations: Stream 2 119 3.1 Week4........................................... 119 3.1.1 Lec 11 Algebraandgeometryasphysics . 119 3.2 Week5........................................... 121 3.2.1 Lec 12 The physics behind complex analytic expressions: linear vs. nonlinear . 121 3.2.2 Lec 13 Rootclassificationofpolynomials . 124 3.2.3 Lec 14 AnalyticGeometry............................... 126 3.3 Week6........................................... 127 3.3.1 Lec 15 Gaussian Elimination of linear equations . 127 3.3.2 Lec 16 Matrixcomposition: BilinearandABCD . 130 3.3.3 Lec 17 Introduction to the Branch cut and Riemann sheets . 131 3.4 Week7........................................... 131 3.4.1 Lec 18 Complex analytic mappings (domain coloring) . 131 3.4.2 Lec 19 Signals and Systems: Fourier vs. Laplace Transforms . 133 3.4.3 Lec 20 Role of Causality and the Laplace Transform . 133 3.5 Week8........................................... 133 3.5.1 Lec 21 The 9 postulates of System of algebraic Networks . 133 3.5.2 Lec 22 ExamII(Evening) ..............................139 CONTENTS 5 4 Scalar Calculus: Stream 3a 141 4.1 Week8........................................... 141 4.1.1 Lec 23 Newton and early calculus & the Bernoulli Family . 141 4.2 Week9........................................... 143 4.2.1 Lec 24 Power series and complex analytic functions . 143 4.2.2 Lec 25 Integrationinthecomplexplane . 143 4.2.3 Lec 26 Cauchy Riemann conditions: Complex-analytic functions . 143 4.3 Week10.......................................... 144 4.3.1 Lec 27 Differentiationinthecomplexplane . 144 4.3.2 Lec 28 ThreecomplexIntegralTheorems . 144 4.3.3 Lec 29 InverseLaplaceTransform. 144 4.4 Week11.......................................... 145 4.4.1 Lec 30 Inverse Laplace Transform & Cauchy residue theorem . 145 4.4.2 Lec 31 Thecaseforcausality ............................. 145 4.4.3 Lec 32 Laplace transform properties: Modulation, time translation, etc. 145 4.5 Week12.......................................... 146 4.5.1 Lec 33 Multi-valued complex functions, Branch Cuts, Extended plane ......146 4.5.2 Lec 34 The Riemann Zeta function ζ(s) .......................146 4.5.3 Lec 35 ExamIII ....................................147 5 Vector Calculus: Stream 3b 149 5.1 Week13.......................................... 149 5.1.1 Lec 36 ScalarWaveequation . 149 5.1.2 Lec 37 Partial differential equations of physics . 149 5.1.3 Lec 38 Gradient, divergence and curl vector operators . 149 5.2 ThanksgivingHoliday ............................. ........150 5.3 Week14.......................................... 150 5.3.1 Lec 39 Geometry of Gradient, divergence and curl vector operators .......150 5.3.2 Lec: 40 IntroductiontoMaxwell’sEquation . 150 5.3.3 Lec: 41 The Fundamental theorem of Vector Calculus . 150 5.4 Week15.......................................... 151 5.4.1 Lec 42: The Quasi-static approximation and applications . 151 5.4.2 Lec 43: Last day of class: Review of Fund Thms of Mathematics . 152 A Notation 155 A.1 Numbersystems ................................... 155 A.1.1 Symbolsandfunctions. 155 A.1.2 Greekletters .................................. 155 A.1.3 Table of double-bold numbernotation ........................ 155 A.2 Complex vectors and the impedance matrix . ............156 A.2.1 Vectors in R3 ......................................157 A.3 Matrices........................................ 158 A.4 Periodicfunctions ............................... ........158 A.5 Differential equations vs. Polynomials . ..............159 B Linear algebra of 2x2 matrices 161 B.1 Notation........................................ 161 B.1.1 Gaussianeliminationexercises . ..........163 B.2 Inverseofthe2x2matrix . ........164 B.2.1 Derivationoftheinverseofa2x2matrix. ...........164 C Eigenvector analysis 165 6 CONTENTS D Solution to Pell’s Equation (N=2) 169 D.1 PellequationforN=3 .............................. .......169 E Laplace transforms 171 F Transmission lines 173 F.0.1 Transferfunctions ............................. 173 G 2D parabolic horn 177 G.0.1 3DConicalHorn ................................. 178 G.0.2 ExponentialHorn ............................... 180 G.1 Derivation of the Webster Horn Equation . ............182 G.2 Theinverseproblem ............................... .......183 G.3 WKBmethod....................................... 183 G.4 Rydbergseries ................................... 184 G.5 Laplacian operator in N dimensions .............................184 H Filter classification 185 H.0.1 Giventhefunction: ............................. 185 H.0.2 Morequestions ................................. 185 H.0.3 Entropyanalysis ............................... 186 I Stillwell’s Intersection vs. composition 187 CONTENTS 7 1 Abstract 2 An understanding of physics requires knowledge of mathematics. The contrary is not true. By defi- 3 nition, pure mathematics contains no physics. Yet historically, mathematics has a rich history filled 4 with physical applications. Mathematics was developed by people with intent of making things work. 5 In my view, as an engineer,
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