Math 3C03 Course Syllabus

Syllabus for Math 3C03 (Mathematical Physics I)
Term1, 2013/14

The following is a tentative syllabus for the course. This page will be updated regularly.
The chapters and sections refer to the text book "Mathematical Methods for Physics and Engineering" by K.F. Riley, M.P. Hobson & S.J. Bence.

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.

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.