Funky Mathematical Physics Concepts

1 Funky Mathematical Physics Concepts The Anti-Textbook* A Work In Progress. See for the latest versions of the Funky Series. Please send me comments. Eric L. Michelsen TijxvxTijyvyTijzvz+dR realimaginaryCICRi-iRCI I study mathematics to learn how to think. I study Physics to have something to think about. Perhaps the greatest irony of all is not that the square root of two is irrational, but that Pythagoras himself was irrational. * Physical, conceptual, geometric, and pictorial Physics that didn t fit in your textbook. Please do NOT distribute this document. Instead, link to Please cite as: Michelsen, Eric L., Funky Mathematical Physics Concepts , , 12/18/2018. Funky Mathematical Physics Concepts emichels at 12/18/2018 2:34 PM Copyright 2002-2018 Eric L.

2 Michelsen. All rights reserved. 2 of 299 2006 values from NIST. For more physical constants, see . Speed of light in vacuum c = 299 792 458 m s 1 (exact) Boltzmann constant k = 6504(24) x 10 23 J K 1 Stefan-Boltzmann constant = 400(40) x 10 8 W m 2 K 4 Relative standard uncertainty x 10 6 Avogadro constant NA, L = 141 79(30) x 1023 mol 1 Relative standard uncertainty x 10 8 Molar gas constant R = 472(15) J mol-1 K-1 Electron mass me = 382 15(45) x 10 31 kg Proton mass mp = 621 637(83) x 10 27 kg Proton/electron mass ratio mp/me = 672 47(80) Elementary charge e = 176 487(40) x 10 19 C Electron g-factor ge = 319 304 3622(15) Proton g-factor gp = 694 713(46) Neutron g-factor gN = 085 45(90) Muon mass m = 531 30(11)

3 X 10 28 kg Inverse fine structure constant 1 = 999 679(94) Planck constant h = 068 96(33) x 10 34 J s Planck constant over 2 = 571 628(53) x 10 34 J s Bohr radius a0 = 177 208 59(36) x 10 10 m Bohr magneton B = 915(23) x 10 26 J T 1 Reviews .. most excellent tensor I feel I have come to a deep and abiding understanding of relativistic The best explanation of tensors seen anywhere! -- Physics graduate student Funky Mathematical Physics Concepts emichels at 12/18/2018 2:34 PM Copyright 2002-2018 Eric L. Michelsen. All rights reserved. 3 of 299 Contents 1 Introduction .. 9 Is There Another Kind of Physics ? .. 9 Why Funky ? .. 9 How to Use This Document .. 9 Why Physicists and Mathematicians Argue.

4 9 Thank You .. 10 Scope .. 10 Notation .. 10 2 Random Short Topics .. 13 I Always Lie .. 13 What s Hyperbolic About Hyperbolic Sine? .. 13 Basic Calculus You May Not Know .. 15 The Product Rule .. 16 Integration By Pictures .. 16 Theoretical Importance of IBP .. 18 Delta Function Surprise: Coordinates Matter .. 18 Spherical Harmonics Are Not Harmonics .. 20 The Binomial Theorem for Negative and Fractional Exponents .. 21 When Does a Divergent Series Converge? .. 22 Algebra Family Tree .. 23 Convoluted Thinking .. 24 3 Vectors .. 26 Small Changes to Vectors .. 26 Why (r, , ) Are Not the Components of a Vector .. 26 Laplacian s Place .. 27 Vector Dot Grad Vector .. 35 4 Green Functions .. 37 The Big Idea .. 37 Boundary Conditions on Green Functions.

5 42 Introduction to Boundary Conditions .. 42 One Dimensional Boundary Conditions .. 43 2D?? and 3D Green Functions .. 49 Green Functions Don t Separate .. 49 Green Units .. 50 Special Case: Laplacian Operator with 3D Boundary Conditions .. 51 Desultory Green Topics .. 54 Fourier Series Method for Green Functions .. 54 Green-Like Methods: The Born Approximation .. 57 5 Complex Analytic Functions .. 59 Residues .. 60 Contour Integrals .. 61 Evaluating Integrals .. 61 Choosing the Right Path: Which Contour? .. 64 Evaluating Infinite Sums .. 69 Multi-valued Functions .. 71 6 Conceptual Linear Algebra .. 73 Matrix Multiplication .. 73 Determinants .. 74 Cramer s Rule .. 75 Area and Volume as a Determinant .. 76 Funky Mathematical Physics Concepts emichels at 12/18/2018 2:34 PM Copyright 2002-2018 Eric L.

6 Michelsen. All rights reserved. 4 of 299 The Jacobian Determinant and Change of Variables .. 77 Expansion by Cofactors .. 79 Proof That the Determinant Is Unique .. 81 Getting Determined .. 82 Advanced Matrices .. 83 Getting to Home Basis .. 83 Diagonalizing a Self-Adjoint Matrix .. 84 Contraction of Matrices .. 86 Trace of a Product of Matrices .. 86 Linear Algebra Briefs .. 87 7 Probability, Statistics, and Data Analysis .. 88 Probability and Random Variables .. 88 Precise Statement of the Question Is Critical .. 89 How to Lie With Statistics .. 90 Choosing Wisely: An Informative Puzzle .. 90 Multiple Events .. 91 Combining Probabilities .. 92 To B, or To Not B? .. 94 Continuous Random Variables and Distributions .. 95 Population and Samples .. 96 Population Variance.

7 97 Population Standard Deviation .. 97 New Random Variables From Old Ones .. 98 Some Distributions Have Infinite Variance, or Infinite Average .. 99 Samples and Parameter Estimation ..100 Why Do We Use Least Squares, and Least Chi-Squared ( 2)? ..100 Average, Variance, and Standard Functions of Random Variables ..104 Statistically Speaking: What Is The Significance of This? ..105 Predictive Power: Another Way to Be Significant, but Not Important ..108 Unbiased vs. Maximum-Likelihood Estimators ..108 Correlation and Dependence ..110 Independent Random Variables are Uncorrelated ..111 r You Serious? ..112 Statistical Analysis Algebra ..113 The Average of a Sum: Easy? ..113 The Average of a Product ..113 Variance of a Sum ..114 Covariance Revisited ..114 Capabilities and Limits of the Sample Variance.

8 114 How to Do Statistical Analysis Wrong, and How to Fix It ..117 Introduction to Data Fitting (Curve Fitting) ..118 Goodness of Fit ..119 Linear Regression ..123 Review of Multiple Linear Regression ..123 We Fit to the Predictors, Not the Independent Variable ..124 The Sum-of-Squares Identity ..126 The Raw Sum-of-Squares Identity ..127 The Geometric View of a Least-Squares Fit ..128 Algebra and Geometry of the Sum-of-Squares Identity ..129 The ANOVA Sum-of-Squares Identity ..130 The Failure of the ANOVA Sum-of-Squares Identity ..131 Subtracting DC Before Analysis ..132 Fitting to Orthonormal Functions ..132 Hypothesis Testing with the Sum of Squares Identity ..132 Funky Mathematical Physics Concepts emichels at 12/18/2018 2:34 PM Copyright 2002-2018 Eric L.

9 Michelsen. All rights reserved. 5 of 299 Introduction to Analysis of Variance (ANOVA) ..133 The Temperature of Liberty ..134 The F-test: The Decider for Zero Mean Gaussian Noise ..137 Coefficient of Determination and Correlation Coefficient ..138 Uncertainty Weighted Data ..141 Be Sure of Your Uncertainty ..141 Average of Uncertainty Weighted Data ..141 Variance and Standard Deviation of Uncertainty Weighted Data ..143 Normalized weights ..145 Numerically Convenient Weights ..146 Transformation to Equivalent Homoskedastic Measurements ..146 Linear Regression with Individual Uncertainties ..148 Linear Regression With Uncertainties and the Sum-of-Squares Identity ..149 Hypothesis Testing a Model in Linear Regression with Uncertainties ..153 8 Practical Considerations for Data Analysis.

10 154 Rules of Thumb ..154 Signal to Noise Ratio (SNR) ..154 Computing SNR From Data ..155 Spectral Method of Estimating SNR ..156 Fitting Models To Histograms (Binned Data) ..157 Reducing the Effect of Noise ..160 Data With a Hard Cutoff: When Zero Just Isn t Enough ..162 Filtering and Data Processing for Equally Spaced Samples ..163 Finite Impulse Response Filters (aka Rolling Filters) and Boxcars ..163 Use Smooth Filters (not Boxcars) ..164 Guidance Counselor: Computer Code to Fit Data ..164 9 Numerical Analysis ..168 Round-Off Error, And How to Reduce It ..168 How To Extend Precision In Sums Without Using Higher Precision Variables ..169 Numerical Integration ..170 Sequences of Real Numbers ..170 Root Finding ..170 Simple Iteration Equation ..170 Newton-Raphson Pseudo-Random Numbers.