Week | Sections in Text | Suggested Homework (not to be handed in) |
Comments |
---|---|---|---|
05/09 to 06/09 |
Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7 |
8.19 Exercises: 8.1, 8.3, 8.4, 8.7, 8.9 |
The first
lecture was introductory. Click here for my synopsis of basic linear algebra, that you need to know. Please read it carefully. |
09/09 to 13/09 |
Chapters 8 and 9 |
8.19 Exercises: 8.10, 8.12, 8.19, 8.20, 8.22, 8.26, 8.31, 8.38, 8.39, 8.42 |
This
week, I reviewed basic facts about linear algebra:
vector spaces (over the
real and complex numbers), linear maps, rank and
nullity, matrices, Gaussian elimination, determinants,
inner product spaces (both real and complex), orthogonal
and unitary transformations, base change,
eigenvalues, eigenvectors, diagonalisation of
matrices (one of the
most crucial aspects of linear algebra!),
decompositions such as LU, QR, SVD, etc. The
material corresponds to Chapter 8 in the textbook. I also
discussed normal
modes of
oscillatory systems, which is an application of
eigenvalue problems to physics.
This is in Chapter 9 in the textbook. |
16/09 to 20/09 |
Chapters 9, 12
and 13.1 ( and 3.1, 3.2, 3.3, 3.4 for a quick review of complex numbers) |
9.4
Exercises: 9.1, 9.2, 9.3, 9.10 12.9 Exercises: 12.4, 12.5, 12.6, 12.8, 12.13, 12.14, 12.15, 12.16, 12.19, 12.20, 12.22, 12.23, 12.24 13.4 Exercises: 13.1, 13.4, 13.5, 13.8, 13.9, 13.10, 13.11, 13.12 |
On Monday, I finished Chapter 9 on normal modes. The
rest of the week will be about Fourier Series, Fourier Transform and the Laplace Transform.
This corresponds to Chapters 12 and 13 in the textbook. (you should also have a look at 3.1, 3.2, 3.3, 3.4 for a quick review of complex numbers) Click here for my short notes about Fourier Series that you need to know, here for a the basic properties of the Fourier Transform and here for the Laplace Transform Assignment #1 is due this week on Thursday at the beginning of the lecture period. |
23/09 to 27/09 |
Chapters 13, 14 and 15 | 13.4
Exercise: 13.18, 13.21, 13.23, 13.24 14.4 Exercises: 14.2, 14.3, 14.7, 14.8, 14.12, 14.13, 14.18, 14.22, 14.25 15.4 Exercises: 15.5, 15.8, 15.10, 15.12, 15.14, 15.29, 15.30, 15.31 |
Monday
and Tuesday was about applications of the Fourier Transform and the Laplace
Transform. This corresponds to Chapter 13 in the textbook. On Thursday, I gave a quick review of what you should know about O.D.E's (most of it you must have seen in a second year course). We will concentrate on the type of ODE's that are relevant for the subsequent sections of this course. The material corresponds to Chapters 14 and 15 in the textbook. |
30/09 to 04/10 | Chapters 16 | 16.6 Exercises: 16.1, 16.2, 16.5, 16.7, 16.9, 16.10, 16.14, 16.15 |
This
week's lectures was on series solutions of ODE's The
material corresponds to Chapter 16 in the textbook. Instead of talking too much about the abstract theory, I concentrated on solving explicit equations such as the Legendre, Hermite, Laguerre and Bessel equations. The Gamma function was also introduced (needed to define Bessel functions!). Assignment #2 is due on Thursday at the beginning of the lecture period. |
07/10 to 11/10 | Chapter 17 Sections 18.1, 18.2 |
17.7 Exercises:
17.2, 17.3, 17.5, 17.6, 17.7, 17.11, 17.14 |
This
week
I explained the algebraic properties (ortogonality etc.)
of eigenvalue problems for linear second order
ordinary differential operators (Sturm-Liouville
type) acting on function spaces. (Chapter 17 in the
textbook) On Thursday, I began the discussion about Legendre polynomials and special functions (Chapter 18). |
14/10 to 18/10 | Sections 18.1, 18.2,
18.3, 18.4, 18.7, 18.8, 18.9 from Chapter 18. |
18.13 Exercises: 18.2, 18.3, 18.4, 18.5, 18.6 | TEST
#1
was held on Tuesday, Oct. 15th from 19:00 to 20:00. The test
covered the material from Chapters 8, 9, 12, 13, 14, 15
and 16 and what I did in my lectures up to and including
the lecture on Tuesday, October 8th. This week we continued Chapter 18 on special functions by discussing Legendre functions, Legendre polynomials, spherical harmonics, Hermite and Laguerre polynomials. Click here for my notes on Legendre, here for Hermite and here for Laguerre polynomials (notes from last year). On Thursday, I introduced you to the most important PDE's in Mathematical Physics: Laplace, Poisson, Heat (Diffusion), Wave and the Schrödinger equation. |
21/10 to 25/10 | Sections 18.5, 18.6, 18.11, 18.12 from
Chapter 18 and Sections 21.1 from Chapter 21 |
18.13 Exercises: 18.7 18.8, 18.9, 18.10, 18.17 18.23 |
This
week's lectures were more on special functions: Chebyshev, Bessel.
We also started solving important PDE's: Laplace,
Poisson, Heat (Diffusion), Wave and
the Schrödinger equations Click here for my notes on some properties of Bessel that we are using (notes from last year). Assignment #3 is due this week at the begiining of the lecture on Thursday. |
28/10 to 30/10 | Sections 21.2, 21.3, 21.4, 21.5 from Chapter 21 |
21.6 Exercises: 21.2, 21.3, 21.4, 21.5, 21.6, 21.9, 21.10, 21.12, 21.13, 21.14, 21.15, 21.18, 21.19, 21.20, 21.21, 21.22 |
This
week I will introduce you to the hypergeometric
equation and the Beta function. I will also
continue Chapter 21, the key chapter for this course,
by explaining the separation of variables (in Cartesian, polar, cylindrical and
spherical coordinates) method to solve the
most important PDE's in two and three dimensions. |
04/11 to 08/11 | 21.4, 21.5 from Chapter 21 and Chapter 20 | 21.6
Exercises: 21.23, 21.24, 21.25,
21.26, 21.27, 21.28 20.8 Exercises: 20.16, 20.17, 20.18 |
This
week, I did a number of important examples from
Chapter 21. The
fundamental solution to Laplace's equation and Green's functions for
the inhomogeneous case was be explained. I also derived
some important Poisson Formulas (double layer
potentials) that we need. I also explained how to use
Fourier transforms to obtain the Heat Kernel and
described the d'Alembert's general solution for the
one-dimensional wave equation solution Assignment #4 was due this week at the beginning of the lecture on Thursday |
11/11 to 15/11 | Chapter 19 |
19.3 Exercises: 19.4, 19.7, 19.8 |
This week,
I will go through Chapter 19, which is basically an
algebraic description of basic Quantum Mechanics,
including the quantum harmonic oscillator.
I will explain angular momentum, spin, Pauli
matrices and the Lie algebra su(2). TEST #2 was held on Tuesday, Nov. 12th from 19:00 to 20:00 . The test covered the material from Chapters 17, 18, 20 and 21. |
18/11 to 22/11 |
Selected material from Chapters 22 |
22.9 Exercises: 22.4, 22.15, 22.16, 22.19, 22.21, 22.26 |
On
Monday and Tuesday, I solved the non-relativistic
Schrödinger equation for the hydrogen atom and
described the SO(4) symmetry On Thursday, I will discuss the Euler-Lagrange equation in the calculus of variations and compute the bending of light in the Schwarzschild metric of General Relativity. Assignment #5 is due this week at the beginning of the lecture on Thursday |
25/11 to 03/12 | Selected material from Chapters
22 and 23 REVIEW |
On Monday, I will explain the Rayleigh-Ritz
principle for eigenvalues. For the rest of the week,
I will review the material and work out a few sample
questions to prepare you for the Final Examination. I will also tie
up some loose ends. The last lecture will be on
Monday, December 2nd. There will be no lecture on Tuesday,
December 3rd, but you should come to my office to pick up
unclaimed assignments and tests. |
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