Math 3D03 Course Syllabus

Syllabus for Math 3D03 (Mathematical Physics II),
Term2, 2014/15

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 Comments

05/01 to 09/01
Review of Chapter 3 and 4.
Sections 24.1, 24.2, 24.3, 24.4, 24.8
3.8 Exercises: 3.10, 3.12, 3.14, 3.16, 3.18, 3.26, 3.28
4.8 Exercises: 4.16, 4.20, 4.23, 4.33, 4.36 After a very brief review on Tuesday of some basic facts about complex numbers, as contained in Chapter 3, and power series as contained in Chapter 4, I explained the exponential finction, its inverse the loagarithm and why we need a branch cut. After that I defined complex differentiation and derived the CAUCHY-RIEMANN EQUATIONS that analytic (or holomorphic) functions satisfy using the fundamental fact that the derivative is complex linear and hence commutes with multiplication by i. I explained some of the profound consequences of these fundamental equations that lie at the foundadtion of complex analysis. On Friday, I explained how one integrates complex functions, or better, one forms f(z)dz along curves and proved the CAUCHY INTEGRAL THEOREM (the mother of all theorems in complex analysis) as a consequence of the C-R equations and Green's Theorem. I also showed you how to integrate dz/z around a closed loop, which is the mother of all residue calculaions!

12/01 to 16/01
Sections 24.8, 24.9, 24.10, 24.11, 24.12, 24.5, 24.6

24.14 Exercises: 24.2, 24.3, 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 24.10

I continued with the Cauchy integral Theorem, derived the RESIDUE THEOREM and discussed the notion of the winding number. The residue theorem is the key technique (contour integrals) that will be used to evaluate (without using Wolfram alpha!) a number of important definite integrals.
I also talked about ZEROS, POLES, ESSENTIAL SINGULARITIES, TAYLOR series, LAURENT series, etc. I derived the CAUCHY INTEGRAL FORMULA and its variants (you just differentiate what you have!). This is another fundamental ingredient in complex analysis.

19/01 to 23/01
Sections 24.11, 24.12, 24.13, 25.3

24.14 Exercises: 24.11, 24.12, 24.13, 24.14, 24.15, 24.16, 24.17, 24.18, 24.19, 24.20, 24.21, 24.22

Assignment #1 is due this week on Tuesday at the beginning of the lecture period.
This week was devoted to various techniques of contour integration. We evaluated some classical integrals such as the Fresnel Integral.

26/01 to 30/01
Sections 25.3, 25.4
25.9 Exercises: 25.6, 25.7, 25.8, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13, 25.14, 25.15
We will also learn how to use contour integrals to
sum interesting infinite series such as Riemann zeta function at even integers (and also at -1) using Bernoulli numbers. That was fun (at least for me!)
We explored more consequences of the Cauchy integral formula and the residue theorem, such as the Mean Value Property, the Maximum Modulus Principle, the Argument Principle and LIOUVILLE's THEOREM. . The argument principle can be used to count the number of zeros and poles inside a contour (Rouché's Theorem).
The fundamental theorem of algebra follows from that.

02/02 to 06/02 Sections 24.7, 25.1, 25.2 25.9 Exercises: 25.2, 25.3, 25.4, 25.5

Assignment #2 is due on Tuesday at the beginning of the lecture period.

On Tuesday and Thursday, we will talk about CONFORMAL MAPPINGS and how to use them to solve physical and engineering problems in Potential Theory and fluid flow in the plane.
On Friday, we will look at Laplace transforms from the complex point of view and derive a formula for the inverse inverse Laplace transform using a Bromwich contour integral.

09/02 to 13/02 Sections 25.5
25.9 Exercises: 25.12, 25.14, 25.15

TEST #1 will be held on Tuesday, February 10th from 19:05 to 19:55 in BSB 137, 138. The test will cover the material that was done in class up to and including the lecture on Tuesday, February 3rd.
On Tuesday I will give a review and finish the discussion about Bromwich contour integrals.
On Thursday and Friday, we will have a (quick) look at Stokes' equation, Airy integrals and WKB methods.
I will give an application of Airy integrals to quantum tunneling.

Next week is Mid-term Recess

23/02 to 27/02 Sections 25.6, 25.7, 25.8

25.9 Exercises: 25.16, 25.17. 25.18, 25.19, 25.20, 25.21, 25.22, 25.23

On Tuesday, I will wrap up complex analysis by giving a brief overview of general steepest descent and stationary phase methods to approximate integrals. These are essential techniques in Quantum Mechanics.

Assignment #3   is due this week on Thursday in class.

We will begin Probabilty on Thursday!
I will define sample space, events, the axioms of probability and simple consequences thereof, such as the exclusion-inclusion principle. The basic combinatorial counting techniques will be explained, including the multinomial formula and some simple applications.

02/03 to 06/03 Sections 30.1, 30.2, 30.3, 30.4, 30.5, 30.6

30.16 Exercises: 30.3, 30.5, 30.6, 30.9, 30.10, 30.12, 30.14, 30.15, 30.16

On Tuesday, I will define the key concepts of conditional probability, independence and Bayes' Formula.
The rest of the week is devoted to RANDOM VARIABLES and their generating functions, in particular the characteristic function which is just our old friend, the Fourier Transform. I will define the fundamental notion of a random variable, its expected value (or mean) and its Variance (or Volatility (sic))

09/03 to 13/03 Sections 30.7, 30.8, 30.9, 30.10, 30.11
30.16 Exercises: 30.18, 30.20, 30.21, 30.24, 30.25, 30.26, 30.30, 30.32

This week I will introduce the most important probability distributions that are commonly used together with their characteristic functions:
discrete: Bernoulli, binomial, negative binomial, multinomial, hypergeometric, Poisson
continuous: Multivariate Normal (Gaussian), exponential, Gamma, Student t, chi-square, Cauchy etc.
I will also define the key notion of joint distributions, independence of random variables, marginal distributions, covariance, correlation and prove some basic formulas, such as the expectation and variance of sums of random variables.

16/03 to 22/03
Sections 30.12, 30.13, 30.14, 30.15
30.16 Exercises: 30.36, 30.37, 30.39, 30.40
Assignment #4 is due this week in class on Tuesday.

On Tuesday I will prove the Central Limit Theorem
On Thursday I will explain Chebyschev's inequality. Jensen's inequality and do some applications of the CLT.
On Friday, I will prove the convolution formula for the pdf of the sum of two random variables and show that the sum of independent Gaussians is a Gaussian, that the sum of independent Gamma's is again a Gamma and that the sum of independent Poisson is again Poisson.

23/03 to 27/03
Sections 31.1, 31.2, 31.3, 31.4, 31.5

31.8 Exercises: 31.4, 31.5, 31.6, 31.7, 31.8, 31.9, 31.10

TEST #2 will be held tentatively on Tuesday, March 24th from 19:00 to 20:00. The test will cover the material that was done in class up to and including the lecture on Tuesday, March 18th.

This week, I will begin Statistics.
I will explain basic estimators in parametric statistics, their consistency, bias and efficiency and Confidence Intervals. We will also study the maximum likelihood estimator and more about the multivariate normal distribution and its important descendants: chi-squared, student-t and the F-distribution, which play a role in Statistics. We will discuss the Method of Least Squares and Regression and what is known as Hypothesis Testing.

30/03 to 03/04

Sections 31.6, 31.7
Extra material

31.8 Exercises: 31.14, 31.15, 31.17, 31.20
Assignment #5 is due this week in class on Tuesday.

This week we will also discuss some goodness of fit tests including the chi-squared test (which is also useful for testing independence) and the non-parametric Kolmogorv-Smirnov test.

06/04 to 08/04 REVIEW

Week	Sections in Text	Suggested Homework	Comments
05/01 to 09/01	Review of Chapter 3 and 4. Sections 24.1, 24.2, 24.3, 24.4, 24.8	3.8 Exercises: 3.10, 3.12, 3.14, 3.16, 3.18, 3.26, 3.28 4.8 Exercises: 4.16, 4.20, 4.23, 4.33, 4.36	After a very brief review on Tuesday of some basic facts about complex numbers, as contained in Chapter 3, and power series as contained in Chapter 4, I explained the exponential finction, its inverse the loagarithm and why we need a branch cut. After that I defined complex differentiation and derived the CAUCHY-RIEMANN EQUATIONS that analytic (or holomorphic) functions satisfy using the fundamental fact that the derivative is complex linear and hence commutes with multiplication by i. I explained some of the profound consequences of these fundamental equations that lie at the foundadtion of complex analysis. On Friday, I explained how one integrates complex functions, or better, one forms f(z)dz along curves and proved the CAUCHY INTEGRAL THEOREM (the mother of all theorems in complex analysis) as a consequence of the C-R equations and Green's Theorem. I also showed you how to integrate dz/z around a closed loop, which is the mother of all residue calculaions!
12/01 to 16/01	Sections 24.8, 24.9, 24.10, 24.11, 24.12, 24.5, 24.6	24.14 Exercises: 24.2, 24.3, 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 24.10	I continued with the Cauchy integral Theorem, derived the RESIDUE THEOREM and discussed the notion of the winding number. The residue theorem is the key technique (contour integrals) that will be used to evaluate (without using Wolfram alpha!) a number of important definite integrals. I also talked about ZEROS, POLES, ESSENTIAL SINGULARITIES, TAYLOR series, LAURENT series, etc. I derived the CAUCHY INTEGRAL FORMULA and its variants (you just differentiate what you have!). This is another fundamental ingredient in complex analysis.
19/01 to 23/01	Sections 24.11, 24.12, 24.13, 25.3	24.14 Exercises: 24.11, 24.12, 24.13, 24.14, 24.15, 24.16, 24.17, 24.18, 24.19, 24.20, 24.21, 24.22	Assignment #1 is due this week on Tuesday at the beginning of the lecture period. This week was devoted to various techniques of contour integration. We evaluated some classical integrals such as the Fresnel Integral.
26/01 to 30/01	Sections 25.3, 25.4	25.9 Exercises: 25.6, 25.7, 25.8, 25.8, 25.9, 25.10, 25.11, 25.12, 25.13, 25.14, 25.15	We will also learn how to use contour integrals to sum interesting infinite series such as Riemann zeta function at even integers *(and also at -1) using Bernoulli numbers.* That was fun (at least for me!) We explored more consequences of the Cauchy integral formula and the residue theorem, such as the Mean Value Property, the Maximum Modulus Principle, the Argument Principle and LIOUVILLE's THEOREM. . The argument principle can be used to count the number of zeros and poles inside a contour (Rouché's Theorem). The fundamental theorem of algebra follows from that.
02/02 to 06/02	Sections 24.7, 25.1, 25.2	25.9 Exercises: 25.2, 25.3, 25.4, 25.5	Assignment #2 is due on Tuesday at the beginning of the lecture period. On Tuesday and Thursday, we will talk about CONFORMAL MAPPINGS and how to use them to solve physical and engineering problems in Potential Theory and fluid flow in the plane. On Friday, we will look at Laplace transforms from the complex point of view and derive a formula for the inverse inverse Laplace transform using a Bromwich contour integral.
09/02 to 13/02	Sections 25.5	25.9 Exercises: 25.12, 25.14, 25.15	TEST #1 will be held on Tuesday, February 10th from 19:05 to 19:55 in BSB 137, 138. The test will cover the material that was done in class up to and including the lecture on Tuesday, February 3rd. On Tuesday I will give a review and finish the discussion about Bromwich contour integrals. On Thursday and Friday, we will have a (quick) look at Stokes' equation, Airy integrals and WKB methods. I will give an application of Airy integrals to quantum tunneling. Next week is Mid-term Recess
23/02 to 27/02	Sections 25.6, 25.7, 25.8	25.9 Exercises: 25.16, 25.17. 25.18, 25.19, 25.20, 25.21, 25.22, 25.23	On Tuesday, I will wrap up complex analysis by giving a brief overview of general steepest descent and stationary phase methods to approximate integrals. These are essential techniques in Quantum Mechanics. Assignment #3 is due this week on Thursday in class. We will begin Probabilty on Thursday! I will define sample space, events, the axioms of probability and simple consequences thereof, such as the exclusion-inclusion principle. The basic combinatorial counting techniques will be explained, including the multinomial formula and some simple applications.
02/03 to 06/03	Sections 30.1, 30.2, 30.3, 30.4, 30.5, 30.6	30.16 Exercises: 30.3, 30.5, 30.6, 30.9, 30.10, 30.12, 30.14, 30.15, 30.16	On Tuesday, I will define the key concepts of conditional probability, independence and Bayes' Formula. The rest of the week is devoted to RANDOM VARIABLES and their generating functions, in particular the characteristic function which is just our old friend, the Fourier Transform. I will define the fundamental notion of a random variable, its expected value (or mean) and its Variance (or Volatility (sic))
09/03 to 13/03	Sections 30.7, 30.8, 30.9, 30.10, 30.11	30.16 Exercises: 30.18, 30.20, 30.21, 30.24, 30.25, 30.26, 30.30, 30.32	This week I will introduce the most important probability distributions that are commonly used together with their characteristic functions: discrete: Bernoulli, binomial, negative binomial, multinomial, hypergeometric, Poisson continuous: Multivariate Normal (Gaussian), exponential, Gamma, Student t, chi-square, Cauchy etc. I will also define the key notion of joint distributions, independence of random variables, marginal distributions, covariance, correlation and prove some basic formulas, such as the expectation and variance of sums of random variables.
16/03 to 22/03	Sections 30.12, 30.13, 30.14, 30.15	30.16 Exercises: 30.36, 30.37, 30.39, 30.40	Assignment #4 is due this week in class on Tuesday. On Tuesday I will prove the Central Limit Theorem On Thursday I will explain Chebyschev's inequality. Jensen's inequality and do some applications of the CLT. On Friday, I will prove the convolution formula for the pdf of the sum of two random variables and show that the sum of independent Gaussians is a Gaussian, that the sum of independent Gamma's is again a Gamma and that the sum of independent Poisson is again Poisson.
23/03 to 27/03	Sections 31.1, 31.2, 31.3, 31.4, 31.5	31.8 Exercises: 31.4, 31.5, 31.6, 31.7, 31.8, 31.9, 31.10	TEST #2 will be held tentatively on Tuesday, March 24th from 19:00 to 20:00. The test will cover the material that was done in class up to and including the lecture on Tuesday, March 18th. This week, I will begin Statistics. I will explain basic estimators in parametric statistics, their consistency, bias and efficiency and Confidence Intervals. We will also study the maximum likelihood estimator and more about the multivariate normal distribution and its important descendants: chi-squared, student-t and the F-distribution, which play a role in Statistics. We will discuss the Method of Least Squares and Regression and what is known as Hypothesis Testing.
30/03 to 03/04	Sections 31.6, 31.7 Extra material	31.8 Exercises: 31.14, 31.15, 31.17, 31.20	Assignment #5 is due this week in class on Tuesday. This week we will also discuss some goodness of fit tests including the chi-squared test (which is also useful for testing independence) and the non-parametric Kolmogorv-Smirnov test.
06/04 to 08/04	REVIEW