Mathematical Methods and Physical Insights: An Integrated Approach

Description

Mathematics instruction is often more effective when presented in a physical context. Schramm uses this insight to help develop students' physical intuition as he guides them through the mathematical methods required to study upper-level physics. Based on the undergraduate Math Methods course he has taught for many years at Occidental College, the text encourages a symbiosis through which the physics illuminates the math, which in turn informs the physics. Appropriate for both classroom and self-study use, the text begins with a review of useful techniques to ensure students are comfortable with prerequisite material. It then moves on to cover vector fields, analytic functions, linear algebra, function spaces, and differential equations. Written in an informal and engaging style, it also includes short supplementary digressions ('By the Ways') as optional boxes showcasing directions in which the math or physics may be explored further. Extensive problems are included throughout, many taking advantage of Mathematica, to test and deepen comprehension. Cover Half-title Title page Copyright information Dedication Contents List of BTWs Preface Part I Things You Just Gotta' Know 1 Prelude: Symbiosis 2 Coordinating Coordinates 2.1 Position-Dependent Basis Vectors 2.2 Scale Factors and Jacobians Problems 3 Complex Numbers 3.1 Representations 3.2 Euler’s Formula and Trigonometry Problems 4 Index Algebra 4.1 Contraction, Dummy Indices, and All That 4.2 Two Special Tensors 4.3 Common Operations and Manipulations 4.4 The Moment of Inertia Tensor Problems 5 Brandishing Binomials 5.1 The Binomial Theorem 5.2 Beyond Binomials Problems 6 Infinite Series 6.1 Tests of Convergence 6.2 Power Series 6.3 Taylor Series Problems 7 Interlude: Orbits in a Central Potential 7.1 The Runge–Lenz Vector 7.2 Orbits in the Complex Plane 7.3 The Anomalies: True, Mean, and Eccentric Problems 8 Ten Integration Techniques and Tricks 8.1 Integration by Parts 8.2 Change of Variables 8.3 Even/Odd 8.4 Products and Powers of Sine & Cosine 8.5 Axial and Spherical Symmetry 8.6 Differentiation with Respect to a Parameter 8.7 Gaussian Integrals 8.8 Completing the Square 8.9 Expansion of the Integrand 8.10 Partial Fractions Problems 9 The Dirac Delta Function 9.1 The Infinite Spike 9.2 Properties of the Delta Function Problems 10 Coda: Statistical Mechanics 10.1 The Partition Function 10.2 The Chemical Potential 10.3 The Ideal Boson Gas Problems Part II The Calculus of Vector Fields 11 Prelude: Visualizing Vector Fields Problems 12 Grad, Div, and Curl 12.1 The Del Operator 12.2 [vec(nabla)] and Vector Identities 12.3 Different Coordinate Systems 12.4 Understanding nabla[sup(2)], [vec(nabla)]·, and [vec(nabla)]× Problems 13 Interlude: Irrotational and Incompressible Problems 14 Integrating Scalar and Vector Fields 14.1 Line Integrals 14.2 Surface Integrals 14.3 Circulation 14.4 Flux Problems 15 The Theorems of Gauss and Stokes 15.1 The Divergence Theorem 15.2 Stokes’ Theorem 15.3 The Fundamental Theorem of Calculus — Revisited 15.4 The Helmholtz Theorem Problems 16 Mostly Maxwell 16.1 Integrating Maxwell 16.2 From Integrals to Derivatives 16.3 The Potentials Problems 17 Coda: Simply Connected Regions 17.1 No Holes Barred? 17.2 A Real Physical Effect 17.3 Single-Valued Problems Part III Calculus in the Complex Plane 18 Prelude: Path Independence in the Complex Plane 18.1 Analytic Functions 18.2 Cauchy’s Integral Formula Problems 19 Series, Singularities, and Branches 19.1 Taylor Series and Analytic Continuation 19.2 Laurent Series 19.3 Multivalued Functions 19.4 The Complex Logarithm Problems 20 Interlude: Conformal Mapping 20.1 Visualizing Maps 20.2 The Complex Potential Problems 21 The Calculus of Residues 21.1 The Residue Theorem 21.2 Integrating Around a Circle 21.3 Integrals Along the Real Axis 21.4 Integration with Branch Cuts 21.5 Integrals with Poles on the Contour 21.6 Series Sums with Residues Problems 22 Coda: Analyticity and Causality 22.1 Acting on Impulse 22.2 Waves on a String 22.3 The Klein–Gordon Propagator Problems Part IV Linear Algebra 23 Prelude: Superposition Problems 24 Vector Space 24.1 Vector Essentials 24.2 Basis Basics 24.3 Kets and Reps Problems 25 The Inner Product 25.1 The Adjoint 25.2 The Schwarz Inequality 25.3 Orthonormality 25.4 Building a Better Basis: Gram–Schmidt 25.5 Completeness 25.6 Matrix Representation of Operators Problems 26 Interlude: Rotations 26.1 Active and Passive Transformations 26.2 What Makes a Rotation a Rotation? 26.3 Improper Orthogonal Matrices: Reflections 26.4 Rotations in [mathbb(R)][sup(3)] 26.5 Rotating Operators: Similarity Transformations 26.6 Generating Rotations Problems 27 The Eigenvalue Problem 27.1 Solving the Eigenvalue Equation 27.2 Normal Matrices 27.3 Diagonalization 27.4 The Generalized Eigenvalue Problem Problems 28 Coda: Normal Modes 28.1 Decoupling Oscillators 28.2 Higher Dimensions Problems Entr'acte: Tensors 29 Cartesian Tensors 29.1 The Principle of Relativity 29.2 Stress and Strain 29.3 The Equivalence Class of Rotations 29.4 Tensors and Pseudotensors 29.5 Tensor Invariants and Invariant Tensors Problems 30 Beyond Cartesian 30.1 A Sheared System 30.2 The Metric 30.3 Upstairs, Downstairs 30.4 Lorentz Tensors 30.5 General Covariance 30.6 Tensor Calculus 30.7 Geodesics, Curvature, and Tangent Planes Problems Part V Orthogonal Functions 31 Prelude: 1 2 3 . . . Infinity 31.1 The Continuum Limit 31.2 An Inner Product of Functions Problems 32 Eponymous Polynomials 32.1 Legendre Polynomials 32.2 Laguerre and Hermite Polynomials 32.3 Generating Functions Problems 33 Fourier Series 33.1 A Basis of Sines and Cosines 33.2 Examples and Applications 33.3 Even and Odd Extensions Problems 34 Convergence and Completeness 34.1 Pointwise and Uniform Convergence 34.2 Parseval’s Theorem Problems 35 Interlude: Beyond the Straight and Narrow 35.1 Fourier Series on a Rectangular Domain 35.2 Expanding on a Disk 35.3 On a Sphere: The Y[sub(ell m)]’s 35.4 From Shell to Ball Problems 36 Fourier Transforms 36.1 From Fourier Sum to Fourier Integral 36.2 Physical Insights 36.3 Complementary Spaces 36.4 A Basis of Plane Waves 36.5 Convolution 36.6 Laplace Transforms Problems 37 Coda: Of Time Intervals and Frequency Bands 37.1 Sampling and Interpolation 37.2 Aliasing Problems Part VI Differential Equations 38 Prelude: First Order First Problems 39 Second-Order ODEs 39.1 Constant Coefficients 39.2 The Wronskian 39.3 Series Solutions 39.4 Legendre and Hermite, Re-revisited Problems 40 Interlude: The Sturm–Liouville Eigenvalue Problem 40.1 Whence Orthogonality? 40.2 The Sturm–Liouville Operator 40.3 Beyond Fourier Problems 41 Partial Differential Equations 41.1 Separating Space and Time 41.2 The Helmholtz Equation 41.3 Boundary Value Problems 41.4 The Drums Problems 42 Green's Functions 42.1 A Unit Source 42.2 The Eigenfunction Expansion 42.3 Going Green in Space and Time 42.4 Green’s Functions and Fourier Transforms Problems 43 Coda: Quantum Scattering 43.1 The Born Approximation 43.2 The Method of Partial Waves Problems Appendix A Curvilinear Coordinates Appendix B Rotations in [mathbb(R)][sup(3)] Appendix C The Bessel Family of Functions References Index