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Funky Mathematical Physics Concepts Physics Concepts The Anti-Textbook* A Work In Progress. See...

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  • Funky Mathematical

    Physics Concepts

    The Anti-Textbook*

    A Work In Progress. See elmichelsen.physics.ucsd.edu/ for the latest versions of the Funky Series.

    Please send me comments.

    Eric L. Michelsen

    Tijxv x

    Tijyv y

    Tijzv z

    + dR

    real

    imaginary

    CI

    CR

    i

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    R

    CI

    “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 elmichelsen.physics.ucsd.edu/FunkyMathPhysics.pdf. Please cite as: Michelsen, Eric L., Funky Mathematical Physics Concepts, elmichelsen.physics.ucsd.edu/, 2/10/2020.

    https://elmichelsen.physics.ucsd.edu/ https://elmichelsen.physics.ucsd.edu/FunkyMathPhysics.pdf https://elmichelsen.physics.ucsd.edu/

  • elmichelsen.physics.ucsd.edu/ Funky Mathematical Physics Concepts emichels at physics.ucsd.edu

    2/10/2020 10:10 AM Copyright 2002-2020 Eric L. Michelsen. All rights reserved. 2 of 303

    2006 values from NIST. For more physical constants, see http://physics.nist.gov/cuu/Constants/ .

    Speed of light in vacuum c = 299 792 458 m s–1 (exact)

    Boltzmann constant k = 1.380 6504(24) x 10–23 J K–1

    Stefan-Boltzmann constant σ = 5.670 400(40) x 10–8 W m–2 K–4

    Relative standard uncertainty ±7.0 x 10–6

    Avogadro constant NA, L = 6.022 141 79(30) x 1023 mol–1

    Relative standard uncertainty ±5.0 x 10–8

    Molar gas constant R = 8.314 472(15) J mol-1 K-1

    Electron mass me = 9.109 382 15(45) x 10–31 kg

    Proton mass mp = 1.672 621 637(83) x 10–27 kg

    Proton/electron mass ratio mp/me = 1836.152 672 47(80)

    Elementary charge e = 1.602 176 487(40) x 10–19 C

    Electron g-factor ge = –2.002 319 304 3622(15)

    Proton g-factor gp = 5.585 694 713(46)

    Neutron g-factor gN = –3.826 085 45(90)

    Muon mass mμ = 1.883 531 30(11) x 10–28 kg

    Inverse fine structure constant  –1 = 137.035 999 679(94)

    Planck constant h = 6.626 068 96(33) x 10–34 J s

    Planck constant over 2π ħ = 1.054 571 628(53) x 10–34 J s

    Bohr radius a0 = 0.529 177 208 59(36) x 10–10 m

    Bohr magneton μB = 927.400 915(23) x 10–26 J T–1

    Reviews

    “... most excellent tensor paper.... I feel I have come to a deep and abiding understanding of relativistic

    tensors.... The best explanation of tensors seen anywhere!” -- physics graduate student

    https://elmichelsen.physics.ucsd.edu/ http://physics.nist.gov/cuu/Constants/

  • elmichelsen.physics.ucsd.edu/ Funky Mathematical Physics Concepts emichels at physics.ucsd.edu

    2/10/2020 10:10 AM Copyright 2002-2020 Eric L. Michelsen. All rights reserved. 3 of 303

    Contents

    1 Introduction ........................................................................................................................................... 9 Mathematical Physics, or Physical Mathematics? ............................................................................ 9 Why Physicists and Mathematicians Argue ..................................................................................... 9 Why Funky? ..................................................................................................................................... 9 How to Use This Document ............................................................................................................. 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 ........................................................................................................20 Delta Function Surprise: Coordinates Matter ..................................................................................20 Spherical Harmonics Are Not Harmonics .......................................................................................22 The Binomial Theorem for Negative and Fractional Exponents .....................................................23 When Does a Divergent Series Converge? .....................................................................................24 Algebra Family Tree .......................................................................................................................25 Convoluted Thinking ......................................................................................................................26 Two Dimensional Convolution: Impulsive Behavior ......................................................................27 Structure Functions .........................................................................................................................28 Correlation Functions ......................................................................................................................29

    3 Vectors ...................................................................................................................................................30 Small Changes to Vectors ...............................................................................................................30 Why (r, θ, ) Are Not the Components of a Vector ........................................................................30 Laplacian’s Place ............................................................................................................................31 Vector Dot Grad Vector ..................................................................................................................39

    4 Green Functions ...................................................................................................................................41 The Big Idea ....................................................................................................................................41 Boundary Conditions on Green Functions ......................................................................................46 Introduction to Boundary Conditions ..............................................................................................46 One Dimensional Boundary Conditions ..........................................................................................47 2D?? and 3D Green Functions ........................................................................................................53 Green Functions Don’t Separate .....................................................................................................53 Green Units .....................................................................................................................................54 Special Case: Laplacian Operator with 3D Boundary Conditions ..................................................55 Desultory Green Topics ..................................................................................................................58 Fourier Series Method for Green Functions ....................................................................................58 Green-Like Methods: The Born Approximation .............................................................................61

    5 Complex Analytic Functions ...............................................................................................................63 Residues ..........................................................................................................................................64 Contour Integrals .............................................................................................................................65 Evaluating Integrals ........................................................................................................................65 Choosing the Right Path: Which Contour? .....................................................................................68 Evaluating Infinite Sums .................................................................................................................73 Multi-valued Functions ............................................................

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