Mathematical Methods and Physical Insights: An Integrated Approach

✍ Scribed by Alec J. Schramm

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 788
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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