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**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

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Calculus of Variations (Math 433)

Assignments

Topics for term paper

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**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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Assignments

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The following table gives an indication of the scope and

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

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The following table gives an indication of the scope and

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

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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, R^{3}, 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

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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

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The following table gives an indication of the scope and

Topics | Sections | weeks |

Curves, polar coordinates, conic sections | 10.1-10.6 | 2 |

Taylor series | 11.8-11.10 | 1 |

Geometry in R^{3}, 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

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The following table gives an indication of the scope and

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

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The following table gives an indication of the scope and

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

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The following table gives an indication of the scope and

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.

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The following table gives an indication of the scope and

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

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Detailed course outline

Midterm Test (2003)

Anna Sierpinska's site for Linear Algebra I

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Detailed course outline

Notes

Class Sessions

Quizzes and Tests

Dates of Tests and Quizzes

Maple on the Web

Mast 235 site of Fred Szabo

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Detailed course outline

Some notes concerning derivatives

Midterm Test (2002)

Midterm Test (2003)

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Assignment 1

Assignment 2

Maple Code

Mathematica Code

Short reading list

Suggestions for projects

Some related programs (under development):

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