Files and Resources for Math Methods (TECP TTVN)

Spring 2007

Here are the notes:

• 07-01-19: Scaling and the Trinity Explosion

• 07-01-22: Homework

• 07-01-22: Lift Coefficient

• 07-01-24: Taylor Series

• 07-01-26: Dealing with small variations

• 07-01-29: Layers and multiple reflections Also includes Homework Assignment 2.

• 07-01-31: Spherical and cylindrical coords 1

• 07-02-02: Spherical and cylindrical coords 2

• 07-02-05: Gradient 1 Includes introduction to path integral and directional derivative.

• 07-02-05: Newton's Second Law from a potential

• 07-02-07: Total derivative, continuous systems Includes Homework assignment 3.

  Here is a link to the Java demo applets for fields, etc.

• 07-02-09: Introduction to divergence.

• 07-02-12: Stability of orbits in higher dimensions. Revised since presentation.

  It turns out the gradient is really a tensor, a 1-form. I am working on a Draft on tensors if you want to look at it.

• 07-02-14: Useful derivation from 6.2 for 6.5, review of cross product for next chapter.Updated since class

• 07-02-16: Curl, vorticity.

• 07-02-19: Theorem of Gauss.

• 07-02-19: Test 1: Due Tuesday, Feb 27.

• 07-02-21: Pre-class draft: Section 8.3, Acoustic representation theorem. This is in "article" rather than "slide" form, better for printing. Shows many of the steps in the section more explicitly.Slightly revised Wed. morning

• 07-02-21: Slides on 8.3: acoustic representation to go with above. Has different commentary, so look at both of these and the text to understand what is going on.

• 07-02-23: Flow of probability. An application of Gauss's law to quantum mechanics

• 07-02-26: Theorem of Stokes. with some applications to Maxwell's equations.

• 07-02-28: Wingtip vortices. How do airplanes fly?

• 07-03-02: Ch 10: Curvature. Introduction to ch. 10. These notes are short, I will elaborate in class. Includes next homework assignment.

• 07-03-05: Ch 10: Shortest distance between two points. These are the notes I went over in class on paper last time.

  Your text uses the same procedure extended to a two-dimensional soap film to introduce the Laplacian in section 10.3. However, I have made another:

• 07-03-05: Intro to Laplace's equation and harmonic functions.

• 07-03-07: Some properties of the Laplacian, average of a harmonic function.

• '07-03-09 (Friday) Since we discussed doing the class differently, I will mainly try homework help on the last assignment today. Our next chapter will be on analytic functions: Ch. 16. Over or at the end of break I will post some article-form help about complex variables and example problems. See the next note:

• My jury duty was cancelled. March 19. We started discussing chapter 16: Analytic functions

• 2007-03-19 You should work through section 16.1 problems b-e. None of these have difficult steps, in particular b and c are easy: just writing out the differential coefficient and taking the given limits gives you the Cauchy-Riemann equations.

• 2007-03-23
We started covering Ch. 17, up through section 17.3 on calculating integrals with the residue theorem. Since the text only does a few examples without much background, there is more below:

• A typeset PDF file of a couple of residue examples: Rational function residue examples.
• How to find residues (3 pages): Page 1, Page 2, Page 3,
• A trig function example, similar to 17.3 problem f,g,h: Simple trig function example.
• Integrating a rational function of sine and cosine using the unit circle. This is kind of an algebra horror, because it does a general case (2 pages): Page 1, Page 2. Not all of the algebra is worked out explicitly in this one, the spots where I wrote check! are missing steps. Snieder does not do an example like this, but it is another major type of integral that can be done with residues.
• Wikipedia has descriptions of two lemmas used to check that the semicircular arcs in contour integrals go to zero: the estimation lemma, also known as the ML-inequality, and Jordan's lemma.

  The estimation lemma is used between problems 17.3 a and b, and Jordan's lemma is basically problem 17.3 f and 17.4 d, although Snieder does not use these names (nor do all texts.)

• Wikibooks has a short summary of Residue theory. It looks good but use with caution.
• Even better: The math department at Cal State Fullerton has a set of Complex variables modules for undergrads that look excellent, particularly chapters 6, 7, and 8 (scroll down there) for what we are doing.

Homework for residues: 070328_HW.pdf I originally had this due Friday, Apr. 6, but I forgot that Easter was this close, so it is due Monday, Apr. 9, since there probably won't be anyone here Friday either. The problems are fairly short and just like the examples, so you can probably get it done early.

Here is a single-sheet: Basic terminology and formulas for complex variables (PDF).

'07-04-02 (Monday)We started on Ch. 15 -- Fourier analysis, covering section 15.1. We also talked about analogies with finite vector spaces.

The fourier series applet is here.

'07-04-04 (Wednesday)We will briefly continue talking about vector and function spaces and the complex form of Fourier series.

We will also go over some concepts and properties of the Dirac delta function. Chapter 14 covers this in detail but we mainly just want results, in particular equations 14.11,17,20,24, and 31.

There is an uncorrected error in Ch. 14: on page 206, bottom paragraph under problem a, the sentence should be: 'This expression states that the "surface area" under the delta frequence is one.'

Here is Test 2,

'07-04-13: Here is some info on Fourier transform conventions for the most common systems, with some pointers to resources on the web. This is a work in progress, covering differences between the various locations of pi, etc.

The Final Exam is here. It is due Wednesday, May 2.

Formularies

Math formulary (PDF)

Physics formulary (PDF)

These are free formula compendia off of the web. The original address is in the introduction to each file. I will probably replace them with links to the original site soon.

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