Course Information
| Differential Geometry I (Math 380) |
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| Multivariable Calculus I (Mast 218) |
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| Multivariable Calculus II (Mast 219) |
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| Advanced Calculus I (Math 264) |
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| Advanced Calculus II (Math 265 |
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| Differential Equations (Math 370) |
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| Partial Differential Equations (Math 371) |
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| Partial Differential Equations (Math 473) (Mast 666) |
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| Complex Analysis (Math 366) |
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| Linear Algebra I (Math 251) |
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| Linear Algebra (Maple) II (Mast 235) |
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| Calculus I (Math 203) |
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| Calculus of Variations (Math 433) |
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| Variational Methods (Mast 689) |
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| Quantum Mechanics (Mast 684) |
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| Numerical Analysis (Mast 680B Mast 683) |
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| Computational Applied Mathematics (Mast 680J) |
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| Spectral Geometry (Mast 855) |
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| Classical Optimization (Math 618) |
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| Reading List |
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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 assignments 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
Back to the topVariational 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.
Assignments
Topics for term paper
Back to the topComputational 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
Back to the topQuantum Mechanics (Mast 684, Math 434J & 856B)
Aim: The course is designed to introduce students of mathematics to quantum mechanics. It will start with a short review of calculus of variations and classical mechanics and then go on to Hilbert space and quantum mechanics. The emphasis will be mostly on non-relativistic bound systems. Students will be expected to solve problems in mechanics, applied functional analysis, and quantum mechanics. The course will include topics from: calculus of variations, classical mechanics, Hilbert space, operator theory, Fourier analysis, analytically soluble QM problems, symmetries, angular momentum, estimation of spectra, variational methods, Rayleigh-Ritz theorem, identical particles, the many-body problem.Prerequisites: Analysis and differential equations.
Evaluation: There will be a number of problem sets, and a short project on an approved topic.
Books: An annotated reading list on mathematics and quantum mechanics may be found on my web page. Students are advised to study a text book such as Mathematical concepts of quantum mechanics by S. J. Gustafson and I. M. Sigal (Springer, 2003 - 2011), or An introduction to quantum theory by K. Hannabuss (Oxford, 1997). It may also be helpful to consult an introductory QM text such as that by Greiner (QM Vol I, Springer, 1989), books on classical mechanics, such as Goldstein (Addison-Wesley, 1980 - 2001), and on operator theory, such as Schechter (North Holland, 1981) or Analysis by Lieb and Loss (AMS, 1997 - 2001).
Assignments
Back to the topDifferential Geometry I (Math 380)
Text Book Elementary Differential Geometry by B. O'Neill (Academic Press, 1997)Evaluation There will be weekly assignments (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 |
Assignments
List of errors in the text book
Tests and Exams
Back to the topAdvanced Calculus II (Math 265)
Text Book Multivariable Calculus by James Stewart (Brooks/Cole, 5th ed 2003)Evaluation There will be weekly assignments (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 |
Assignments
Sample Test and Exam
Back to the topMultivariable Calculus I (Mast 218)
Text Book Multivariable Calculus by James Stewart (Thomson [Brooks/Cole], 8th edition)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A + ,(0.3)T + (0.6)E or (0.1)A + (0.9)E.
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 |
| Parametrically defined plane curves, polar coordinates, conic sections | 10.1-10.6 | 3 |
| Taylor series review, R3, vectors, equations of lines and planes, quadric surfaces | 11.10, 12.1-12.6 | 2 |
| Vector functions, space curves, curvature, particle motion | 13.1-13.4 | 2 |
| Partial derivatives, tangent planes, gradient, linear approximations, maxima and minima | 14.1-14.8 | 4 |
Assignments
Books
Solutions and Notes
Back to the topMultivariable Calculus II (Mast 219)
Text Book Multivariable Calculus by James Stewart (Thomson [Brooks/Cole], 8th edition)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A + ,(0.3)T + (0.6)E or (0.1)A + (0.9)E.
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 | 15.1 - 15.9 | 7 |
| Vector calculus | 16.1 - 16.9 | 6 |
Assignments
Books
Solutions, notes, detailed outline
Back to the topAdvanced Calculus I (Math 264)
Text Book Multivariable Calculus by James Stewart (Thomson [Brooks/Cole], 7th edition)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A +(0.3)T +(0.6)E or (0.1)A + (0.9)E.
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 |
| Curves, polar coordinates, conic sections | 10.1-10.6 | 2 |
| Taylor series | 11.8-11.10 | 1 |
| Geometry in R3, vectors, lines, planes, quadric surfaces | 12.1-12.6 | 2 |
| Vector functions, curves, particle motion | 13.1-13.4 | 2 |
| Continuity, partial derivatives, tangent planes, linear approximations, chain rule, directional derivatives, gradient | 14.1-14.6 | 3 |
| Maxima and minima, Lagrange multipliers | 14.7-14.8 | 2 |
Assignments
Books
Solutions and Notes
Back to the topDifferential Equations (Math 370)
Text Book Elementary Differential equations and boundary value problems by W.E.Boyce and R.C.DiPrima, Edition 10 (Wiley, 2012 & 2013)Evaluation There will be weekly assignments (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 weights will be: assignments (10%) and final exam (90%).
The following table gives an indication of the scope and approximate pace of the course, in terms of time spent on sections of the text book.
| Topics | Sections | number of weeks on topic |
| Introduction | 1.1-1.3 | 1 |
| First-order differential equations | 2.1-2.6 | 3 |
| Existence of solutions. Numerical methods | 2.7-2.8 | 1 |
| Second-order differential equations | 3.1-3.8 | 3 |
| Higher-order differential equations | 4.1-4.4 | 2 |
| Solutions as power series | 5.1-5.3 {+ ... ?} | 2 |
Assignments
Assignments and solutions
Back to the topPartial Differential Equations (Math 371)
Text Book Elementary Differential equations and boundary value problems by W. E. Boyce and R. C. DiPrima (Ed. 9, Wiley, 2009)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A +(0.3)T +(0.6)E or (0.1)A + (0.9)E.
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 | Number of weeks on topic |
| Introduction: 1st order equations; characteristics | 10.1 | 1 |
| Derivation of heat flow and wave equations | 10: Appendices A and B | 1 |
| pde's and boundary-value problems | 11.1 | 1 |
| Fourier Series | 10.2 - 10.4 | 2 |
| Solution of the heat equation | 10.5 - 10.6 | 2 |
| Solution of the wave equation | 10.7 | 1 |
| Laplace's equation | 10.8 | 1 |
| Sturm-Liouville problems | 11.2 - 11.4 | 2 |
Assignments
Books
Solutions and Notes
Back to the topPartial Differential Equations (Math 473) (Mast 666)
Text Book Basic Partial Differential Equations by David Bleecker and George Csordas (International Press, Somerville, 1996 or later)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A +(0.3)T +(0.6)E or (0.1)A + (0.9)E.
The following table gives an indication of the scope and approximate pace of the course, in terms of chapters of the text book.
| Topic | Chapters | Number of weeks on topic |
| Introduction: 1st order equations | 1 & 2 | 1 |
| Fourier Series | 4 | 2 |
| Heat and diffusion equation | 3 | 3 |
| Sturm-Liouville theory | 4 | 2 |
| Wave equation | 5 | 3 |
| Laplace's equation | 6 | 1 |
Assignments
Books
Solutions, Notes, tests, etc.
Back to the topComplex Analysis (Math 366)
Text Book Complex Variables and Applications by J. W. Brown and R. V. Churchill (Edition 8, McGraw-Hill, 2009)Evaluation There will be weekly assignments A, one class test T, and a final examination E. A missed assignment or test cannot be made up. The final grade will be based on the larger of (0.1)A +(0.3)T +(0.6)E or (0.1)A + (0.9)E.
The following table gives an indication of the scope and approximate pace of the course, in terms of chapters of the text book.
| Topics | Chapters | Number of weeks on topic |
| Introduction | 1 | 1 |
| Analytic functions | 2 | 2 |
| Elementary functions | 3 | 2 |
| Complex integration | 4 | 2 |
| Taylor and Laurent series | 5 | 2 |
| Residue theorem and applications | 6 & 7 | 2 |
| Selected topics | 8 & 9 & 12 | 1 |
Assignments
Books
Solutions and Notes
Back to the topLinear Algebra I (Math 251)
Evaluation There will be one class test (30%), and a final examination (70%). A missed test cannot be made up. If it benefits the student, the weight of the final exam will be increased to 100%.
Detailed course outline
Midterm Test (2003)
Anna Sierpinska's site for Linear Algebra I
Back to the topLinear Algebra (Maple) II (Mast 235)
Detailed course outline
Notes
Class Sessions
Quizzes and Tests
Dates of Tests and Quizzes
Maple on the Web
Mast 235 site of Fred Szabo
Back to the topCalculus I (Math 203)
Text Book Single Variable Calculus by James Stewart (Brooks/Cole, 2003)Evaluation There will be weekly assignments (10%), one class test (15%), and a final examination (75%). 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%.
Detailed course outline
Some notes concerning derivatives
Midterm Test (2002)
Midterm Test (2003)
Back to the topClassical Optimization (Math 618)
Course Outline
Assignment 1
Assignment 2
Maple Code
Mathematica Code
Short reading list
Suggestions for projectsSome related programs (under development):
de A program for solving first and second order ordinary differential equations [de.exe]
cvt Calculus of Variations: (i) Input F(x,y,y') and y" = f(x,y,y') (ii) Input trial function y(x; a) [cvt.exe]
cvn Calculus of Variations, Numerical solution: Input F(x,y,y') [cvn.exe]
nmg Nelder-Mead Simplex Algorithm for Minima [nmg.exe]
Back to the topSpectral Geometry (Mast 855)
References A list of suitable books and papers on mathematics and quantum mechanics will be given out. Prerequisites This course has three aspects: mathematics, quantum mechanics, and approximation theory. The techniques are mostly from real analysis. Previous experience with quantum mechanics and operator theory would be helpful, so would some knowledge of computing.Evaluation There will be assignments (50%) and a small project (50%).
Description Geometrical considerations of the type we shall study in this course enter a quantum-mechanical spectral problem when the Hamiltonian operator H = -Delta + V(x,v), which acts on a suitable domain in L2(Rn), depends smoothly on certain real parameters v. Spectral objects such as eigenvalues depend in turn on v and in so doing they generate curves or surfaces in a Euclidean space. It is the goal of spectral geometry to study these manifolds. Reference to established results from functional analysis is needed to set up the basic problem after which standard techniques of real analysis become the principal tools. For a full appreciation of the significance of these efforts some knowledge of quantum mechanics would be very useful. Since exact results are not always possible, the course will explore what might be called spectral approximation theory. Students will be encouraged to use numerical and graphical methods to investigate specific problems. Access to a micro-computer laboratory will be available, including a set of elementary graphics procedures in C++ for those who wish to use them. Solutions obtained with the aid of Mathematica or Maple are also welcome. Some computer work will be unavoidable.
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