PHYS4112 INTERMEDIATE MATHEMATICAL PHYSICS

FALL 2009

CLASS: 10:30AM–12:00N T-TH, Nicholson 435

Prof. Jonathan P. Dowling

Office Hours: 12:00N–1:00PM T-Th

Required Book:
Mathematical Methods in the Physical Sciences, Third Edition
by
Mary Boas
ISBN 0471198269, 9780471198260

Required Software: Acrobat Reader 9.1 & Mathematica 7.01 (Available in PAWS  Tigerware)

NRL Plasma Formulary

100% Homework

SYLLABUS

Chapter 10 Tensor Analysis
Chapter 11 Special Functions
Chapter 12 Legendre, Bessel, Hermite, and Laguerre functions
Chapter 13 Partial Differential Equations

HOMEWORK         ASSIGNED         DUE                 HINTS                        SOLUTIONS
HW01                       THU 27 AUG       THU 03 SEP             HW01HINTS
HW02                       THU 03 SEP        THU 10 SEP
HW03                       THU 10 SEP        THU 17 SEP
HW04                       THU 17 SEP       THU 24 SEP
HW05                       THU 24 SEP        THU 08 OCT
HW06                       THU 08 OCT       THU 15 OCT
HW07                       THU 15 OCT       THU 22 OCT
HW08                       THU 22 OCT        THU 29 OCT
HW09                       THU 29 OCT       THU 05 NOV
HW10                       THU 05 NOV       THU 12 NOV
HW11                       THU 12 NOV       THU 19 NOV
HW12                       THU 19 NOV       THU 10 DEC

LECTURES (PDF) & MATHEMATICA NOTEBOOKS (NB)

Ch.10      Tensor Analysis
10.1           Introduction to Tensors
10.2           Definition of 3-D Cartesian Tensors
10.3           Einstein Summation Convention
10.4           Moment of Inertia Tensor / Mathematica Notebook: 10.4.nb
10.4.A       Appendix: Review of Eigenvalues and Eigenvectors
10.5           Kronecker Delta and Levi-Civita Symbols / Mathematica Notebook: 10.5.nb
10.6           Pseudo-Vectors and Pseudo-Tensors
10.7           More Applications: Moment of Inertia Revisited
10.8           Curvilinear Coordinates
10.9           Vector Operators — Grad, Div, Curl, and Laplacian — in Curvilinear Coordinates
10.10         Non-Orthogonal Curvilinear Coordinates: Covariant and Contravariant Tensors

Ch.11        Special Functions
11.1             Speed Integration by Parts
11.2             Integral form of the Factorial Function
11.3             Euler Gamma Function as Generalized Factorial Function and the Recursion Relation
11.4             Euler Gamma Function of Negative Numbers
11.5             Special Values of Euler Gamma and the Reflection Formula
11.6             The Euler Beta Function
11.7             Euler Beta in terms of Euler Gamma
11.8             Exact Solution to the Simple Pendulum with Beta Function
11.9             Gaussian Integrals and the Error Functions Erf, Erfc, and Erfi
11.10           Asymptotic Series Expansions with Applications to ArcTan and Erf
11.11           Asymptotic Series Expansion of the Euler Gamma Function and Sterling's Approximation for n!
11.9-11        Mathematica Notebook for Above Sections [NB, PDF]
11.12           Jacobi Elliptic Integrals and Functions [NB, PDF]
11.13           Euler-MacLaurin & Abel-Plana Summation Formulas [Dowling89d, Prellberg07]

Ch.12        Series Solutions to Differential Equations
12.1             Introduction
12.2             Legendre's Differential Equation and the Legendre Polynomials
12.3             Leibnitz Product Rule
12.4             Rodriques Formula for Legendre Polynomials
12.5             Generating Function for Legendre Polynomials [NB, PDF]
12.6             Complete Sets of Orthonormal Functions
12.7             Orthogonality of the Legendre Polynomials
12.8             Normalization of the Legendre Polynomials
12.9             Completeness of the Legendre Polynomials
12.10           Associated Legendre Functions
12.11           Generalized Power Series and the Method of Frobenius
12.12           Bessel's Differential Equation and the Bessel Function of the first kind [NB, PDF]
12.13           Bessel's Function of the Second Kind
12.14           Graphs and Zeros of the Bessel's Functions
12.15           Recursion Relations
12.16           General Differential Equations with Bessel Function Solutions
12.17           Other Kinds of Bessel Functions (Airy, Kelvin, SphericalHankel, Modified-Hyperbolic)
12.18          The Bouncing Quantum Ball [NB, PDF]
12.19          The Orthogonality of Bessel's Functions
12.20          Asymptotic Forms of Bessel's Functions [NB, PDF]
12.21          Fuch's Theorem
12.22          Hermite and Laguerre Polynomials

Ch.13         Partial Differential Equations
13.1            Introduction
13.2            Laplace Equation in 2D Cartesian Coordinates: Temperature of a Semi-Infinite Plate
13.2            Mathematica Notebook [NB, PDF]
13.3           Time-Dependent Schrödinger Equation Solution to 2D Particle in a Box
13.3            Mathematica Notebook [NB, PDF]
13.4            Vibrating String
13.4            Mathematica Notebook [NB, PDF]
13.5            Laplace Equation in 3D Cylindrical Coordinates: Temperature of a Semi-Infinite Cylinder
13.6            Vibration of a Circular Membrane: The Drum-Head Problem
13.7            Laplace's Equation in 3D Spherical Coordinates: Temperature of a Solid Sphere

SPECIAL EVENTS
TUE 24 AUG:   FIRST DAY OF CLASS
THU 01 OCT:   FALL HOLIDAY
THU 26 NOV:  THANKSGIVING HOLIDAY
THU 03 DEC:   LAST DAY OF CLASS

SUPPLEMENTAL BOOKS

Entire Course:
http://www.amazon.com/Schaums-Outline-Mathematical-Handbook-Formulas/dp/0070602247
http://www.amazon.com/Schaums-Outline-Mathematica-Eugene-Don/dp/007135719X

Ch.10:

http://www.amazon.com/Schaums-Outline-Vector-Analysis-Spiegel/dp/007060228X

Ch.11

Ch.12:

http://www.amazon.com/Schaums-Outline-Differential-Equations-Outlines/dp/0071456872

Ch.13:

http://www.amazon.com/Schaums-Outline-Partial-Differential-Equations/dp/0070178976

Ch.14:

http://www.amazon.com/Schaums-Outlines-Variables-Introduction-Applications/dp/0070602301

Ch. 15:

http://www.amazon.com/Schaums-Outline-Probability-Statistics-Spiegel/dp/0071350047