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
Grader: Kaushik
Seshadreesan
Required Book:
Mathematical Methods in the Physical
Sciences, Third Edition
by
Mary Boas
Published by Wiley, 2005
ISBN 0471198269, 9780471198260
Required Software: Acrobat Reader 9.1
&
Mathematica 7.01 (Available in PAWS Tigerware)
NRL Plasma Formulary
Dyadic Tensor
Notation
COURSE GRADE:
100% HomeworkSYLLABUS
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
HW01SOL
HW02
THU 03
SEP THU 10 SEP
HW02SOL
HW03
THU 10
SEP THU 17 SEP
HW03SOL
HW04
THU 17 SEP
THU 24 SEP
HW04SOL
HW05
THU 24 SEP
THU 08 OCT
HW05SOL
HW06
THU 08 OCT THU 15
OCT
HW06SOL
HW07
THU 15 OCT THU 22
OCT
HW07SOL
HW08
THU 22 OCT THU
29 OCT
HW08SOL
HW09
THU 29 OCT THU 05 NOV
HW09SOL
HW10
THU 05 NOV THU
12 NOV
HW10SOL
HW11
THU 12
NOV THU 19 NOV
HW11SOL
HW12
THU 19
NOV THU 10 DEC
HW12SOL
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, Spherical,
Hankel,
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-Advanced-Mathematics-Engineers-Scientists/dp/0070602166
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
http://www.amazon.com/Schaums-Advanced-Mathematics-Engineers-Scientists/dp/0070602166
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