Courses

Course Information

	Differential Geometry I (Math 380)
	Advanced Calculus II (Math 265)
	Multivariable Calculus I (Mast 218)
	Differential Equations (Math 370, Mast 330)
	Linear Algebra I (Math 251)
	Linear Algebra (Maple) II (Mast 235)
	Calculus I (Math 203)
	Calculus of Variations (Math 433)
	Variational Methods (Mast 689)
	Quantum Mechanics (Mast 684)
	Numerical Analysis (Mast 680B Mast 683)
	Computational Applied Mathematics (Mast 680J)
	Spectral Geometry (Mast 855)
	Classical Optimization (Math 618 E)
	Reading List

Last update 12 November 2006

Numerical Analysis in C++ (Mast 680B)

Numerical Analysis (Mast 683)

References Students will need to consult books on Numerical Analysis and Scientific Computing. NR = Numerical Recipes by Press et al (Cambridge, 1986) and Introducing C++ for Scientists, Engineers, and Mathematicians by Capper (Springer, 1994) are recommended. NR is well described by its title: for mathematical proofs and detailed results, one must look at the references, and other mathematical literature.
Prerequisites Undergraduate analysis, differential equations, and linear algebra. The C++ language itself will be introduced and studied as part of this course.
Evaluation There will be a sequence of assigments gradually increasing in difficulty. In the final assignment, which will have the flavour of a small project, students will choose one problem from a list of about 10.
Aims This is an elementary course in numerical analysis and computing. Various fundamental topics in numerical analysis will be included. There will be a bias towards analytical problems involving roots, integration, differential equations, optimization, and Fourier transforms. The use of `functional programming' and graphical techniques will be strongly encouraged. Students will be shown how to extend an initial C++ graphics class so that they can take advantage of graphical methods for exploratory purposes. By the end of this course students should have made a good start on the construction of a personal library of tools for exploring and solving mathematical problems numerically.
Computing Students registered in the course will be able to receive a computer account from H925. At the University, the supported dialects are Turbo C++ and Visual C++. These are available in PC LAB C (H923 and H925-1). Students are strongly encouraged to use more private computing environments to which they may have convenient access. For graphics the best dialects for the course are those such as Turbo and Visual that are based on 'Windows'. NB Students who are new to computing should expect that this course will occupy much more time than an ordinary course of mathematics.

Examples and Assignments
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Variational Methods (Mast 689)
Calculus of Variations (Math 433)

Text A reading list will be provided, but the principal text for the course is Calculus of Variations by I. M. Gelfand and V. S. Fomin (Prentice-Hall, 1963; Dover 2000 [ISBN 0486414485])
Prerequisites analysis, differential equations
Evaluation There will be assignments and a substantial term paper. Students will consult with the professor concerning topics for their term papers.
Topics Variational problems, function spaces, Fréchet and Gâteaux variations of a functional, necessary conditions for an extremum, lemmas of Dubois-Reymond, Euler's equation, n unknown functions, higher derivatives, variable end points, coner conditions, side conditions and constraints, canonical form, classical dynamics, Noether's theorem, invariance, symmetries, Lie groups, sufficient conditions for extrema, Weierstrass's E-function, Hamilton-Jacobi theory, problems involving multiple integrals, continuum mechanics, quantum mechanics, direct methods, dynamic programming.

Computational Applied Mathematics (Mast 680J)

References Students will need to consult books on various topics in mathematics and computing. An annotated reading list is accessible from the RH home page: http://www.mathstat.concordia.ca/faculty/rhall/
Prerequisites analysis, differential equations, numerical analysis. Students who take this course are expected to have some familiarity with at least one high-level computing environment such as Maple, Mathematica, or Matlab. If there is sufficient interest, some time may be devoted to programming directly in C++.
Evaluation There will be two projects, chosen in consulatation with the professor. The first one will be selected from a list of suggestions provided; the second will be individually chosen and more ambitious. They will each require detailed design and planning, both mathematically and computationally.
Aims In the initial lectures, mathematical and computational aspects of some problems chosen from the following areas will be considered: approximation theory, ordinary differential equations, calculus of variations, dynamical systems, partial differential equations, integral equations, Fourier transforms, Sturm-Liouville problems, Monte Carlo methods, discrete spectra of Schrödinger operators. By arrangement, each student will choose a project area; the style of the course will then become that of a workshop. It is a principal aim of the course that the students generate well-documented programs that solve mathematical problems.

Assignments
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Quantum Mechanics (Mast 684)

Books An annotated reading list on mathematics and quantum mechanics may be obtained from the RH web page. Students are advised to acquire a copy of a text book such as Mathematical concepts of quantum mechanics by S. J. Gustafson and I. M. Sigal (Springer, 2003, 2006 Softcover), or An introduction to quantum theory by K. Hannabuss (Oxford, 1997). It may also be necessary to consult an introductory QM text such as that by Greiner (Springer, 1989), books on classical mechanics, such as Goldstein (Addison-Wesley, 1980), and on operator theory, such as Schechter (North Holland, 1981) or Analysis by Lieb and Loss (AMS, 1997).
Prerequisites Analysis and differential equations.
Evaluation There will be a number of problem sets.
Aim To introduce mathematics students to quantum mechanics. The course will start with classical mechanics and then go on to the quantum mechanics of non-relativistic bound systems. Students will be expected to solve problems in mechanics, applied functional analysis, and quantum mechanics. The course will include: Classical mechanics, variational principles, Hamilton-Jacobi theory, Hilbert spaces, operator theory, Fourier analysis, elementary soluble problems in QM, angular momementum, symmetry, estimation of spectra, Rayleigh-Ritz theorem, variational methods, identical particles, many-body problem.

Differential Geometry I (Math 380)

Text Book Elementary Differential Geometry by B. O'Neill (Academic Press, 1997)
Evaluation There will be weekly assigments (10%), one class test (30%), and a final examination (60%). A missed assignment or test cannot be made up. If it benefits the student, the weight of the final exam will be increased to 100%.
The following table gives an indication of the scope and approximate pace of the course, in terms of sections of the text book. Students are expected to read Chapter 3 on Euclidean Geometry the subject matter of which will be examined.

Topics	Sections	weeks
Introduction: a first look at some important concepts	1.1-1.7	2
Curves and Frame fields: the Serret-Frenet apparatus and some extensions	2.1-2.8	3
Test on Chapters I and II	-	1
Euclidean geometry: essentially to be read by the student	3.1-3.5	1
Calculus on a manifold: differentiation, integration, and Stokes's theorem	4.1-4.6	6

Advanced Calculus II (Math 265)

Text Book Multivariable Calculus by James Stewart (Brooks/Cole, 5th ed 2003)
Evaluation There will be weekly assigments (10%), one class test (20%), and a final examination (70%). A missed assignment or test cannot be made up. If it benefits the student, the weight of the final exam will be increased to 100%.

The following table gives an indication of the scope and approximate pace of the course, in terms of sections of the text book.

Topics	Sections	weeks
Multiple integrals	16.1 - 16.9	4
Vector calculus	17.1 - 17.9	5
Second-order differential equations	18.1 - 18.4	2

Multivariable Calculus I (Mast 218)

Text Book Multivariable Calculus by James Stewart (Thomson [Brooks/Cole], 6th edition)
Evaluation There will be weekly assigments (10%), one class test (30%), and a final examination (60%). A missed assignment or test cannot be made up. If it benefits the student, the weight of the final exam will automatically be increased to 100%.

The following table gives an indication of the scope and approximate pace of the course, in terms of sections of the text book.

Topics	Sections	weeks
Sequencies and series	12.1-12.11	4
Polar coordinates	11.3	1
Review of geometry in R³, vectors	13.1-13.6	1
Vector functions, curves, particle motion	14.1-14.4	2
Partial derivatives, extrema	15.1-15.8	4

Differential Equations (Math 370, Mast 330)

Text Book Elementary Differential equations and boundary value problems by W.E.Boyce and R.C.DiPrima (Wiley, 2005)
Evaluation There will be weekly assigments (10%), one class test (30%), and a final examination (60%). A missed assignment or test cannot be made up. If it benefits the student, the weight of the final exam will be increased to 100%.
The following table gives an indication of the scope and approximate pace of the course, in terms of sections of the text book.

Topics	Sections	weeks
Introduction	1.1-1.4	1
First-order differential equations	2.1-2.7	3
Second-order differential equations	3.1-3.9	3
Higher-order differential equations	4.1-4.4	2
Laplace Transforms	6.1-6.6	3