Detailed Outline

Mathematical Methods in Physics (PHY 230) - Fall 2006

This is for the detailed outline after the classes. The shorter outline has the full term.

Numbers in parentheses are the lectures (n) or the Computer Labs (Ln). Numbers in square brackets [] are the McQuarrie sections [n.m] or problems [n.m.p], or handouts [Hn], or overheads [On].

In a few places there are extra references beyond the book (McQuarrie) and the course. You don't need to read this, but maybe it's useful. The extra references are:

Strogatz, Nonlinear Dynamics and Chaos
RHB1 = Riley, Hobson, and Bence, Edition 1, Mathematical Methods...
RHB2 = Riley, Hobson, and Bence, Edition 2, Mathematical Methods...

(See the Bibliography).

Introduction (1) [O1]
Special Functions (1-3)
- Elementary functions (1) [1.1]
- Principal values (P or PP) (1) [H1]
- Error functions (erf, erfc) (1) [3.3, H1]
- Gaussian integrals (1)
  - Standard Gaussian integral (int e^-t^2 dt = sqrt(pi)) [3.1.10]
  - Trick: Dimensional analysis
  - Trick: Differentiate under the integral sign [3.4.11]
- Double factorials (1-2) [3.1.15, 3.1.16]
- Gamma function (2-3) [3.1, H1, H2, O2]
  - Euler integral definition (2)
  - Recurrence relation (2)
  - Factorials (2)
  - Gamma(n/2) (2)
  - Extending the definition to negative x (2)
  - Complex z and basic complex numbers (3) [Chap 4]
  - Reflection formula, duplication formula (2)
  - Digamma & polygamma functions and Euler's constant (2)
  - Stirling's Approximation and "Steepest Descents" (2)
- Beta function (2) [3.2, H1]
- Exponential integrals (3) [3.4, H1]
- Elliptic integrals (3) [3.5, H1]
- Riemann zeta function (3) [2.3, 3.7, H1]
- Delta functions (3) [H3]
  - Discrete and continuous delta "functions" [3.6]
  - Delta sequences [3.6]
  - Integral represention
  - Transformed of variables
  - Dimensions
Infinite Series (3-5)
- Convergence and uniform convergence (3-4) [1.2, 2.2, 2.5, L2]
- Convergence tests (4) [2.3, 2.4, H5]
  - Ratio test
  - Liebnitz' alternating series test
  - Absolute convergence
- Expanding Series (4)
  - Standand series:
    - Geometric series [2.2, H4]
    - Binomial series [2.7, H4]
    - Exponential series [H4]
    - Log series [2.6, H4]
    - Riemann zeta series [2.3, 3.7, H1]
  - Taylor/Maclaurin series/expansion [2.7-8]
  - By integration [2.6, 2.8, H4]
  - Bernoulli numbers [3.7, H4]
- Summing series (4-5)
  - Known series! (4)
  - Transform to known series (substitution, add/subtract, partial fractions, differentiatate, integrate, generalize) (4)
  - Rational sums (4) [H6]
  - Trigonometrical series (5)
ODEs (5-9)
- Definitions/Terminology (5) [11, H7]
- General ODEs
  - First order ODEs (5) [11.1-2, H7]
    - Separable [H7#1]
    - Exact [H7#2]
    - Integrating factor [H7#3]
    - Linear eqn [H7#4]
    - Bernoulli's eqn [H7#5]
    - x=F(dy/dx,y) (5) [H7#6]
    - y=F(dy/dx,x) (5) [H7#7]
  - Second order ODEs (6) [11.5, H7]
    - Initial-value and boundary-value problems
    - No y [H7#8]
    - No x [H7#9]
  - Other general ODEs (6)
    - Miscellanous substitutions [H7#10-12, 14-16]
    - Homogeneous/scale-invariant/isobaric eqn [H7#10-12]
- Linear ODEs (6-7) [11.3-6, H7]
  - General solutions (complementary and particular) (6) [H7#18]
  - Complementary solutions (6)
    - Constant-coefficient equations [11.3, H7#19]
    - Euler-Cauchy eqn [11.5, H7#21]
    - Legendre eqn [H7#22]
    - Reduction of Order [11.3, H7#24]
  - Particular solutions (6)
    - Undetermined coefficients [11.4, H7#20]
    - Variation of parameters [11.4, H7#25]
  - Linear independence and Wronskians (7) [9.5, 11.3]
  - Abel's formula (7) [11.3.23, H7#26]
  - Systems of simultaneous ODEs (7) [11.6]
  - Important ODEs and functions (7) [12.3-6, 14.1-2, 16.6, H8]
- Series solutions (7-9) [12]
  - Legendre's ODE (7) [12.3, H9]
  - Legendre polynomials and functions (8) [12.3, H9]
  - Convergence and Fuchs' Theorems (8) [2.6, 18.5, 12.2-4, H10]
  - Classification (OP, RSP, ISP) (8) [12.2, H10]
  - Frobenius method - RSP (8) [12.4, H10]
  - Bessel's ODE (8) [12.5]
  - Bessel functions - J_m(x) (9) [12.5-6, H11]
  - Second solution (9) [12.4, H7#26]
  - Bessel functionis - N_m(x) (9) [12.5-6, H11, O4]
  - x₀ --> infinity (9)

--- End of Test 1 material ---
--- Start of Test 2 material ---

Qualitative Methods (9-10) [13]
- Indroduction (9)
- 1d flows (9, L4) [Strogatz, 13.5, O5]
- 2d flows - fixed points and closed orbits (9, L4) [13.1, 13.4, O5]
- Linear stability analysis (10) [13.2-3]
- Poincaré theorem - fixed points and classification (10) [13.3, O6]
- 3d flows - fixed points, closed orbits, and strange attractors (10)
- Lorenz Equations and chaos (10) [13.5, O7]

Orthogonal Functions and Eigenfunctions (10-13) [14]
- Orthogonal sets of functions (10) [14.2, H12, H13]
- Orthogonal series and generalized Fourier coefficients (10) [14.1-2, 14.4]
- Example: Legendre Series (11) [H9, O8]
- Convergence: (11) [14.1-2, 14.4]
  - Best approximations
  - Bessel's inequality
  - Parseval's equality
  - Convergence in the mean
  - Completeness
- Closure (11)
- Sturm-Liouville theory (12-13)
  - Sturm-Liouville form, regularity conditions, suitable BCs (12) [14.3, H14]
  - Eigenvalues and eigenfunctions (examples) (12) [14.4]
  - Integrating factor --> Sturm-Liouville form (12) [14.3]
  - Suitable boundary conditions, and regular, periodic, singular (12) [14.3, H14]
  - Complex vector spaces and Hermitean operators (12) [9.7, 14.3, H15]
  - Hermitean operators --> real eigenvectors and orthogonal eigenvectors (13) [10.2, 14.3, H15]
- Gram-Schmidt orthogonalization (13) [14.2, H15]
- Legendre Polynomials (13, Legendre-F06.nb/MMA)
  - Explicit polynomials (MMA) [14.1]
  - Rodrigues' formula (MMA) [H9]
  - Recurrence relations (MMA) [14.1, H9]
  - Generating function (13) [14.1, H9]
  - Legendre and Chebyshev series (11, O8, L5, MMA)
Green's Functions (13-14)
- Introduction (13-14)
- Matching method (14) [14.5, H16, RHB1 13.2.5/RHB2 15.2.5]
- Using Green's functions (example) (14) [14.5, H16]
- Eigenfunction method (14, L6) [14.5, H16, RHB1 15.6-7/RHB2 17.6-7]
- 3D Green's functions - Poisson equation (14, L6) [H17, RHB1 17.5.0-3/RHB2 19.5.0-3]
Orthogonal Coordinates (15-16)
- Cylindrical, spherical, and general coordinates (15) [8.3-5, H18-20]
- Coordinate curves and coordinate surfaces (15) [8.6, H18-19]
- Base vectors, unit vectors, and scale factors (15) [8.3-6, H18-20]
- Jacobian matrix and metric tensor (15) [H18-20]
- Infinitesimal displacment, arc length, volume, Jacobian (15) [8.3-6, H18-20]
- Vector components (16) [8.2, H18-20]
- Vector operators (grad, del, curl, Laplacian) (16) [8.3-8.5, H18-20]
PDEs (16-19)
- PDEs of physics (16-17)
  - Classification and boundary conditions (16) [16.7, H21]
  - Laplace and Poisson equations (16) [16.1-2]
  - Heat/diffusion equation (16) [16.1, 16.5]
  - Wave equation (16) [16.1, 16.3-4]
  - Schrödinger equations (16) [16.1, 16.6]
  - Klein-Gordon equation (16)
  - Bending beam equation (17) [17.3]
  - Navier-Stokes equation (17)
  - Reaction-Diffusion equations (17)
- Techniques for solving PDEs (17)
- Separation of variables (SofV) + Superpositions (17-19)
  - Laplace's equation - Cartesian coordinates - SofV (17) [16.2, H22]
  - Laplace's equation - Cylindrical coordinates - SofV (17) [16.4, H11, H18, H22, O10]
    - Modified Bessel ODEs and functions - I_m(x) and K_m(x) (17) [12.5-6, 16.2, H8, H23, O10]
  - Laplace's equation - Cartesian coordinates - Cubical box (18, L7) [16.2]
  - Laplace's equation - Cylindrical coordinates - Cylinder (18) [16.2]

--- End of Test 2 material ---

Wave and heat/diffusion equations (18) [16.3-6]
Helmholtz' equation (18) [16.1, H24]
Schrödinger equation with V(r)=0 (19) [16.6]
Laplace vs. Helmholtz (different BCs, initial conditions) (19) [H21]
Helmholtz' equation - Spherical coordinates (19) [16.2, H19, H24]
- Associated Legendre Functions (19) [16.6, SphericalHarm-F06.nb, O11]
- Spherical Bessel functions (19) [see p. 616 and section 16.8!, O10]
- Spherical Harmonics (19-20) [16.6, SphericalHarm-F06.nb, O11]
- Laplace series (20)

Integral Transforms (20-23)
- General transforms (function/sequences <--> function/sequences) (20) [17.0, H26]
- Fourier series and orthogonal functions (20) [15, H25, H12]
  - Trigonometric/Complex/Cosine/Sine Fourier Series [15.1-2]
  - Periodic and non-periodic [15.2]
  - Parseval's Theorem [15.3]
  - Integration term-by-term [15.3]
  - Useful relations -- Riemann zeta function [15.3]
  - Using ODE [15.4]
- Fourier transforms (20-22)
  - Definition [17.5, H27]
  - From Fourier series to Fourier transforms [17.5]
  - Fourier integral theorem (20) [17.5]
  - Convergence (20) [17.5]
  - Notations (20)
  - Gaussian and uncertainty principle (21) [17.5, H27]
  - Square pulse, sinc(x) (21) [17.5, H27]
  - f(t)=1 (21) [H3, H27]
  - (Real & even) from/to (real & even) (21)
  - Tables - time shifting (21) [H27]
  - ODEs and PDEs (21) [17.6, H27]
  - Parseval's relation [17.5, H27]
  - 2D and 3D Fourier Transforms (21) [17.5]
  - Convolution (21-22)
    - Definition (21) [17.5, H27]
    - Instrumental resolution (22) [FFT-F06.nb]
    - Transfer function (22) [FFT-F06.nb]
    - Convolution theorem (22) [17.5, H27]
  - Power spectrum density (PSD) (22, L8)
    - Definition (22)
    - Total power and Parseval's relation (22)
    - Discrete and continuous spectra (L8)
    - log-log -- often "scale free" -- Brownian and White noise (L8)
  - Correlation and Wiener-Khinchin (22) [H27]
    - Auto- and cross-correlation definitions
    - Correlation theorem
    - Wiener-Khinchin theorem
- General transforms (22, H26)
- Laplace transforms (22-23)
  - Definitions (22) [17.1, 17.7, 19.1, H26, H28]
  - Fourier and Laplace Transforms (22) [H29]
  - ODEs (22) [17.2-3, H2]
  - PDEs (23) [17.4, H28]
Functions of a Complex Variable (23-27)
- Complex numbers and complex functions (Read) [4, H30, Complex-F06.nb]
- Multivalued functions, branch cuts, and branch points (23) [4.6, 18.1, H30]
- Limits and continuity (23) [18.1, H30]
- Differentiation (23) [18.2, H30]
  - Cauchy-Riemann relations
  - Analytic functions
- Power series (23-24) [18.5, H30]
- Singularities (24) [18.2, H30]
  - Poles and Laurent series
  - Removable singularities
  - Isolated essential singularities
  - Non-isolated singularities---branch points, accumulate poles, other
  - Point at infinity
- Complex integration (24) [18.3]
- Cauchy's theorem (24) [18.3, H30]
- Integrals around poles (24) [18.6]
- Residue theorem (24-25) [18.6, H30, H31]
- Definite integrals (25-26)
  - Arc at infinity, and ML lemma (25) [19.2, H31]
  - Finding residues (25) [18.6, H31]
  - Jordan's lemma (25) [H31]
  - Sinusoidal functions (25) [19.2, H31]
  - Principal values (26) [H31]
  - Related paths (26) [H31]
  - Log trick and branch cuts (26) [H31]
  - Inverse Laplace transform (26) [19.1]
- Cauchy's integral formula (26) [18.4, H30]
- Laurent series (26-27) [18.5, H30]
- Conformal mapping (L9) [19.5-7]
Probability (27-30)
- Permutations and combinations (27) [H32]
- Binomial coefficients (27) [H33]
- Basic probability (27) [21.1, H34]
- Random variables (27) [21.1]
- Probability distributions (27-29) [21.1-2]
  - Cumulative Distribution Function (cdf) (27) [21.1, H35]
  - Definitions (pmf and pdf) and important probability distributions (28) [H35, H36]
  - Mean, <x>, E[X], <f(x)>, moments, variance, sd (28) [21.1-2, H35]
  - Transforms of probability distributions (two ways) (28)
  - Joint distributions (28) [21.1-2, H35]
  - Sums of random variables - convolution (28)
  - Covariance and correlations (29) [21.1-2, H35]
- Binomial distribution (29) [21.1, H36]
- Random walks (29) [21.1.12]
- Random events (29-30)
  - Poisson processes/events/noise (29) [21.5]
  - Poisson distribution (29) [21.1, 21.5, H36]
  - Waiting times (29-30) [21.5]
    - Exponential distribution (29) [H36]
    - Gamma distribution (30) [H36]
- Characteristic functions (30) [21.3, H37]
- Sums of random variables - convolution (30) [21.3, H37]
- Central limit theorem (30) [21.3]