This is the website for the class SIO203B Methods of Applied Analysis.
Time: Tuesdays and Thursdays 9:30-11:00 AM
Room: (Geophysics portion) IGPP Board Room
Room: (PO portion) NTV classroom
This class is the second element in a three-Quarter sequence of mathematical methods for physical scientists, particularly those at Scripps Institution of Oceanography and in Engineering. In the Winter Quarter the class will be taught jointly by Prof Stefan Llewellyn-Smith of Engineering and Prof Bob Parker (the author of this website), in an unusual arrangement: up until mid term the class will be split in two, one division for the physical oceanography students (taught by Prof Llewellyn-Smith), the other for geophysics students, (taught by Prof Parker). At the halfway point the two classes merge into one and come under the instruction of Prof Llewellyn-Smith. During the first half Prof Llewellyn-Smith will be discussing various mathematical methods for solving ordinary and partial differential equations, methods of great utility to workers in PO, but of less interest to geophysicists, while Prof Parker will address linear algebra in its analytic and numerical aspects, together with some other topics in numerical analysis which appear more valuable in the solid Earth sciences. The merged class will cover advanced complex analysis and Fourier theory.
The first few weeks concern complex variable theory. An understanding of the complex plane and its utility in the solution of a variety of problems, particularly the evaluation of integrals, is almost essential for any kind of modern applied mathematics.
Next we will move on to linear algebra, a subject surely familiar to most students. However, emphasis will be on the practical application of linear algebra and its implementation through numerical analysis. We will also discuss optimization with the goal of understanding the method of conjugate gradients for the iterative solution of large linear (and nonlinear) systems.
Then comes another topic of numerical anaylsis, numerical integration, with particular focus on how to perform singular integrals that arise often in analytic solutions of physical problems.
We haven't found a satisfactory text yet; Prof Glenn Ierley is writing a book, which undoubtedly will be a hot best seller. Until then I will post notes with references on this site. The notes will be in PostScript form, which you can download and print. Your browser can be configured to screen PostScript, but most are not set up to do this. There will be homework about once a week; it will form a partial basis for the grade - there will also be a final examination. The homework problem sets (and later, their solutions) will always be posted on this site.
There will be some longer calculations and we expect you to have a computer account and some familiarity with matlab, which will be the computational tool for this class and most others. If you cannot get an account, one will be arranged, but please see your advisor first, since he or she will almost certainly want you to have a computer account for your research.
This is the class website for the geophysics portion; its URL is
http://mahi.ucsd.edu/parker/SIO203B/sio203b.html
The class introduction is just the printout of the initial webpage.
The page will be updated, with homework and solutions and
other information.
Golub, G. H., and Van Loan, C. F., Matrix Computations. The bible of numerical matrix caculations. Too concise for bedtime reading, but an invalubale resource.
Riley, K.F., Hobson, M.P., and Bence, S.J., Mathematical methods for physics and engineering. All the applied mathematics you should know (and more), starting from a relatively elementary base. Complex analysis and linear algebra are covered. As I write this the SIO library does not own a copy of this important book - I have asked them to buy one. My copy is on loan to one of the first-year students.
Complex Analysis 0. Motivation 1. Complex Numbers and Simple Functions 2. Differentiation and Analytic Functions 3. Singularities, Branch Points and Cuts 4. Odds and Ends 5. Integration in the Complex Plane
Linear Algebra 0. The Abstract Linear Vector Space 1. Essential Linear Algebra 2. Matrix Norms 3. QR Factorization 4. Cholesky Factorization and Gaussian Elimination 5. Eigenvalue Systems 6. Updating Linear Systems 7. Numerical Optimization 8. Conjugate Gradients
Numerical Integration 1. Summing and Integrating 2. A Classical Result 3. Mirabile Dictu 4. Complex Enlightenment 5. The Wagging Tail 6. Faster, faster 7. Continued Fractions and Analytic Continuation 8. The Pade' Table 9. The Oscillating Tail 10. An Alternative for Alternating Series
The Geophysics portion of the class is now complete. The notes and homework sets have been erased from the Website.