Quick Links:
Instructor:
Ralf Wittenberg
K-10536, 291-4792
ralf@sfu.ca |
Lectures:
M, W, F 8:30-9:20am
in K 9500 |
Text:
Davis & Snider,
Introduction to Vector Analysis
Wm.C. Brown |
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Announcements:
- Homework Set 11, due by the beginning of the final exam, April 19
Note: I have indicated exactly which questions (2, 3, 6(a,b)) you should hand in and will be marked; have a look at the others for your exam preparation. Also, I will give bonus marks for Maple calculations on question 4. In computing the overall homework grade, I will drop the lowest score, so if you would like to drop this score, this homework is optional; otherwise, you can use it to improve your overall homework grade.
For the exam, I would like you to understand the idea of bases, inner products and orthogonality; how to use the Gram-Schmidt process to obtain an orthogonal basis (as done in class and on HW11), how to compute coefficients of an expansion or approximation in terms of an orthogonal basis, and to compute some simple expansions in Legendre or Fourier bases, as in class and on the homework.
Maple code: Legendre, Taylor and Fourier Approximation
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- Final Exam:
Tuesday, April 19, 12:00-3:00pm, WMC 3520
- Material covered on exam:
All sections of the text by Davis and Snider (except Sections 4.10, 5.3, 5.6, 5.7, 5.8) and also material on Orthogonality
Notes:
Sections 4.5, 5.1, 5.5: no proofs
Section 5.2: only first two Green's formulas (integration by parts)
- Lecture notes for the last week of classes are found in Library Reserves
(You are responsible for the material covered in the 'Orthogonality' notes)
Some references for the material on Legendre and Fourier polynomials are given on the Week 13 page
- Formula sheet for final exam
- Review sessions for final exam:
1. Thursday, April 14, 4:30-6:20pm, AQ 3005
2. Monday, April 18, 4:30-6:20pm, AQ 4150
Come with your questions...
- There are no tutorial sessions this week, as it is already the beginning of the exam period
There will be TA office hours at the usual Thursday time
Marked homework sets (HW 10, and all old sets that have not been picked up) are available in a box outside my office, K-10536
- Spring 2004 Final Exam
Solution on tutorial website
Question 6, on transport theorems, is not relevant to this semester's course
Old final exam for review, with solutions by John Hebron
- Lecture 35: 4 April
Orthogonality, Gram-Schmidt Process, Legendre Polynomials:
General inner product, orthogonality, projection; Gram-Schmidt orthogonalization to construct an orthogonal set from an independent set; inner products for functions, construction of Legendre polynomials by the Gram-Schmidt process
Reading: Lecture notes 'Orthogonality' on library reserves
Additional references: You should be able to find additional information on inner products in any text on linear algebra, and on orthogonal polynomials in a text on numerical analysis.
In particular, section 8.2 (and 8.3) of the text by Burden and Faires (on reserve as the textbook for MACM 316) have information on orthogonal polynomials, the Gram-Schmidt process and Legendre polynomials, as well as on trigonometric polynomials and Fourier series.
Legendre polynomials come up in many contexts in differential equations (especially partial differential equations), and it is difficult to find references that do not proceed from the DE context. You might try some of the following web pages (especially the first):
Notes on special functions - these notes approach the material in a similar way to our approach (just read the first few sections)
Web examples: Legendre - good graphs of the first few Legendre polynomials on [-1,1], and a section on expansion in these polynomials
Legendre Polynomials - a general overview
Orthogonal polynomial - summary of the main orthogonal polynomials
- Lecture 36: 6 April
Approximation by Legendre Polynomials, Trigonometric Polynomials:
Approximation of functions by Legendre polynomials, comparison with Taylor polynomials; trigonometric approximation, orthogonality of sines/cosines, Fourier coefficients
Reading: Lecture notes 'Orthogonality' on library reserves
Additional references: There are many texts that discuss Fourier series, for instance any introductory book on partial differential equations; see for instance Chapter 1 (esp. Section 1.1) of the text by Powers, which is the text for MATH 314 on reserve. [Such texts also give you an indication of how much more there is to learn on the subject...]
If you have the MATH 310 textbook by Boyce and DiPrima, Chapter 10 (Section 10.2) also gives you an introduction.
A web search on Fourier Series yields many results, for instance:
Fourier Series from MathWorld - a good introduction
Fourier Series Approximation - a nice java applet displaying successive Fourier approximations of some functions
Another applet on Fourier series
- Lecture 37: 8 April
Fourier Series, Harmonics, Energy Spectrum:
Maple exampls of approximation of functions by Legendre and trigonometric polynomials; Example: Fourier series of square wave; Fourier series as decomposition of a function into its frequency components, energy spectrum
Reading: Lecture notes on library reserves
Additional references: See above for text and web references; Maple code
- Extra lecture:
Tuesday, April 12, 2:30-4:00pm (depending on student interest), AQ 4150
In this (completely optional) lecture I would like to go over the remaining advanced vector calculus material that was not covered in class, but for which you now have all the background; specifically, transport theorems (Section 4.10), Green's identities, the solution of Poisson's equation, delta functions, and the Fundamental Theorem of Vector Calculus (Sections 5.2-3).
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