Physics 834: Mathematical Methods Autumn, 2011

URL: http://www.physics.ohio-state.edu/~ntg/834/834.php

• 04-Dec-2011 --- Here is the formula sheet for the final exam (which will be given to you).
• 03-Dec-2011 --- *** For the final exam, you will be given formulas such as those from the Jackson covers, for Fourier coefficients, the Sturm-Liouville form, and any integrals you need (except for our favorite contour integrals). ***
• 02-Dec-2011 --- Solutions for PS#10 are available below.
• 30-Nov-2011 --- Lecture notes for today are available below under Class notes. The MATLAB codes used today are laplace_relax_random.m, laplace_relax.m, laplace_relax_pbc.m, and eqheat.m.
• 26-Nov-2011 --- A handout with brief notes on solving PDE's by finite difference methods and integral equations with linear algebra and gaussian quadrature, which we'll touch on in the final two lectures, is linked below under Handouts.
• 23-Nov-2011 --- Added spherical harmonics, gaussian quadrature, and Green's function (under differential equations) example notebooks to the Mathematica examples page.
• 22-Nov-2011 --- An interesting paper with three views on the number of fundamental constants is linked below under Handouts.

Assignments

Required Readings (before the due date!)

21-Sep-2011 Lea, chapter 1 excerpt review of vector calculus (and matrices)
21-Sep-2011 Arfken, chapter 1 plus 2.1-2.5 summary of vector calculus
26-Sep-2011 Lea, chapter 2 excerpt survey of complex analysis
26-Sep-2011 Arfken, sections 6.1-6.6 introduction to complex analysis
28-Sep-2011 Arfken, sections 7.1 and 7.2 evaluating contour integrals; dispersion relations
05-Oct-2011 Arfken, sections 9.4 and 9.5 Singular points of a differential equation and series solutions (Frobenius' method)
05-Oct-2011 Lea, chapter 3 excerpt differential equations (particularly Frobenius method and asymptotic solutions)
12-Oct-2011 Lea, chapter 4 excerpt Fourier series
12-Oct-2011 Arfken, chapter 14 Fourier series
24-Oct-2011 Lea, chapter 6 excerpt Generalized functions in physics
24-Oct-2011 Arfken, section 1.15 Fourier series
02-Nov-2011 Lea, chapter 7 excerpt Fourier transforms
02-Nov-2011 Arfken, sections 15.1-15.7 Fourier transforms
09-Nov-2011 Lea, chapter 8 excerpt Sturm-Liouville theory
09-Nov-2011 Arfken, chapter 10 Sturm-Liouville theory
23-Nov-2011 Lea, Appendix C excerpt Green's functions
23-Nov-2011 Arfken, Sections 9.7 and 10.5 Green's functions
TBA Lea, chapter 5 excerpt Laplace transforms

Problem Sets and Hints

Click on the problem set number to get a pdf copy of the assignment.

28-Sep-2011 #1 Hints, suggestions, etc. solutions
05-Oct-2011 #2 Hints, suggestions, etc. solutions
12-Oct-2011 #3 Hints, suggestions, etc. solutions
17-Oct-2011 #4 Hints, suggestions, etc. solutions
26-Oct-2011 #5 Hints, suggestions, etc. solutions and notebook
03-Nov-2011 #6 Hints, suggestions, etc. solutions and notebook
09-Nov-2011 #7 Hints, suggestions, etc. solutions and notebook
17-Nov-2011 #8 Hints, suggestions, etc. solutions and notebook
23-Nov-2011 #9 Hints, suggestions, etc. solutions and notebook
01-Dec-2011 #10 Hints, suggestions, etc. solutions and notebook

Handouts

PDF Copies of Handouts

13-Sep-2011 Jackson covers The inside covers of J.D. Jackson's "Classical Electrodynamics" text with vector calculus stuff.
18-Sep-2011 Using δ's and ε's Summary and examples of how to use the Kronecker delta function and the Levi-Civita (epsilon) symbol.
18-Sep-2011 Excerpts from Wigner article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner (1960).
12-Oct-2011 Excerpts from Blatt article "Practical Points Concerning the Solution of the Schr\"odinger Equation" by John M. Blatt (1967).
04-Nov-2011 Excerpt from Hassani text Defining a delta function at the origin in spherical coordinates (i.e., ignorable coordinates and so on).
14-Nov-2011 Excerpt from Landau QM-II text Conventions for three-dimensional quantum mechanics.
16-Nov-2011 Excerpt from Jackson 7.10 Causality and Kramers-Kronig dispersion relation.
21-Nov-2011 Trilogue on # of constants Three articles by Lev Okun, Gabriele Veneziano, and Michael Duff, who have differing views on the number of fundamental dimensional constants in physics.
26-Nov-2011 PDE's numerically Brief notes on solving PDE's by finite difference methods and integral equations with linear algebra and gaussian quadrature.

Class Notes

The notes are in PDF format only.

21-Sep-2011 lecture 1 Vector calculus review
26-Sep-2011 lecture 2 Vector calculus follow-ups plus first part of complex variables/functions
28-Sep-2011 lecture 3 More on complex variables/functions and complex integration
03-Oct-2011 lecture 4 Contour integrals
05-Oct-2011 lecture 5 More contour integrals and start of series solutions for differential equations.
10-Oct-2011 lecture 6 Loose ends on complex analysis and Frobenius method and asymptotic solutions for differential equations.
14-Oct-2011 lecture 7 Midterm notes, differential equations in Mathematica, gamma function, numerical solutions of diff eqs., Fourier series (pass 1)
17-Oct-2011 lecture 8 Midterm basic examples, gamma function follow-ups, Fourier series (pass 2) with Mathematica notebook on Fourier series (the original notebook is on the Mathematica examples page)
24-Oct-2011 lecture 9 Midterm comments, continuation of Fourier series with new Mathematica notebook on Fourier series
26-Oct-2011 lecture 10 Follow-ups on Fourier series with (yet another) new Mathematica notebook on Fourier series; overview of generalized functions.
31-Oct-2011 lecture 11 More on delta functions including a Mathematica notebook on delta functions.
02-Nov-2011 lecture 12 Wrap-up on delta functions plus first look at Fourier transforms..
07-Nov-2011 lecture 13 More on Fourier transforms and applications. Two Mathematica notebooks: one on defining simplication rules and another with an animation of the solution to a diffusion problem.
09-Nov-2011 lecture 14 Loose ends on Fourier transforms and introduction to Sturm-Liouville theory.
14-Nov-2011 lecture 15 More on Sturm-Liouville theory.
16-Nov-2011 lecture 16 Still more on Sturm-Liouville theory.
21-Nov-2011 lecture 17 Tail end of Sturm-Liouville theory plus Gaussian quadrature.
23-Nov-2011 lecture 18 Comments on PS#8 and PS#9, introduction to Green's functions.
28-Nov-2011 lecture 19 Continuation of Green's functions. Numerical solutions to PDE's and integral equations.
30-Nov-2011 lecture 20 Loose ends, comments on PS#9 and PS#10, emphasis points for final exam.

Mathematical Methods References

Especially Recommended

The call number is linked to the OSCAR library entry. In some cases, this entry includes the table of contents of the book. Some of these will be on reserve at the library.

M. L. Boas Mathematical methods in the physical sciences QA37.2 .B59 2006 Excellent undergraduate text. This is probably your best choice for more basic (or at least alternative) introductions to the topics we will study in 834. Good problems as well.
M. R. Spiegel Schaum's Outline of Advanced Mathematics for Engineers and Scientists TA333 .S6 1971eb The link is to an E-book version. This is a reasonable source of concise summaries of formulas but even better as a source of mostly worked out problems and extra problems with answers (but not solutions). The table of contents matches the 834 syllabus rather well. Costs less than \$15 on Amazon.com.
J. J. Kelly Graduate mathematical physics: with MATHEMATICA supplements QC20.K42 2006 Text for a one-semester course similar to 834, with integrated examples for using Mathematica. Comes with a Mathematica CD-ROM.
M. Stone and P. Goldbart Mathematics for physics: a guided tour for graduate students QC20.S76 2009 The link is to the E-book.
R. Courant and D.Hilbert Methods of Mathematical Physics (2 volumes) QC20 .C851 1953 Classic reference that has everything you need, but not so easy. Available in a new printing (but it is not cheap!). The library has the second volume on differential equations as an e-book.
P. M. Morse and H. Feshbach Methods of Theoretical Physics (2 volumes) QC20.M67 1953 Another classic with some things you won't find elsewhere. From the 1950's but now back in print at Feshbach Publishing.

Other Useful References

M. Abramowitz and I. A. Stegun Handbook of mathematical functions online version Great source of formulas for special functions and other useful things. You can download a pdf version here (but the quality is not so good).
R. V. Churchill and J. W. Brown Complex Variables and Applications QA331.7 .C524 2009 Recommended reference for the theory of analytic functions.
A. L. Fetter and J. D. Walecka Theoretical Mechanics of Particles and Continua QA808.2 .F47 Has good treatments of the general string equation, Sturm-Liouville problems, Green's functions, and more.
I. S. Gradshteyn and I. M. Ryzkhik Table of Integrals, Series, and Products QA55.G6613 2007eb Still indispensible, although Mathematica can replace a lot of it. The link is to an e-book.
S. Hassani Mathematical methods for students of physics and related fields QC20.H394 2009eb Advanced undergraduate level. The link is to the E-book.
J. D. Jackson Classical Electrodynamics QC631.J3 1999 Treats boundary-value problems of electromagnetism and other topics such as dispersion theory.
M. J. Lighthill Introduction to Fourier Analysis and Generalised Functions QA404.L73 A great little book on Fourier transforms.
J. Mathews and R. L. Walker Mathematical Methods of Physics QA401.M29 1970 Another classic text, full of useful "tricks".
H. M. Schey Div, Grad, Curl, and all that QA433.S28 2005 Very accessible ("informal") text on vector calculus.
Sal Mangano Mathematica Cookbook Safari E-book with many Mathematica examples.