Mathematics (MATH)

Students develop their understanding of the essential functions of mathematics (lines and quadratics, polynomial and rational functions, exponential and logarithmic functions, trigonometric functions) and are provided with extensive contextual practice with algebraic manipulation, equation solving, and curve sketching. Students focus on those topics/skills that are known to be essential for subsequent study of calculus. Classes 3 hrs. plus recitation 1 1/2 hrs. per week.

This course is intended for those who are teaching or who plan to teach at the elementary school level. Fundamental concepts and topics in mathematics provide the necessary background for elementary school teachers. Topics include: the axioms of arithmetic and algebra, the integer, rational, and real systems, sets and logic. Classes 3 hrs. and recitation 1.5 hrs. a week.

This course is designed for those who teach or who plan to teach at the junior high or high school level. Topics include: algebra, geometry, probability, and statistics. Material is taught in significantly greater depth than would actually be taught in the school classroom. Classes 3 hrs. a week plus recitations 1.5 hrs. a week

This course is an introduction to Euclidean geometry, designed for those who teach or who plan to teach mathematics at the junior high or high school level. Topics include: coordinate, axiomatic, constructive, and transformational geometry with some emphasis on the concept of mathematical proof.

This is a first course in calculus, intended for students in the CMSE program. The course begins with an overview of essential notions concerning real numbers and functions (such as limits, continuity, sequences and series) before proceeding with a detailed study of the derivative and various applications (including rates of change, approximation, and optimization).

This is a continuation of MATH 1208 and is intended for students in the CMSE program. The course focuses on the development of the Riemann integral and its applications. Core topics include the Fundamental Theorem of Calculus, techniques of integration, computation of areas/volumes, first-order differential equations, and power series.

This is a first course in calculus, intended for science and engineering students. Core topics include: functions, limits, continuity, differentiability; derivatives of algebraic and transcendental functions, applications of the derivative (e.g., curve sketching, optimization, L’Hôpital’s Rule); antiderivatives; area under curves; and the fundamental theorem of calculus. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

This is a continuation of MATH 1210, and is intended for science and engineering students. Core topics include: techniques of integration, applications of the definite integral (e.g. area, volume, arc length); improper integrals, separable differential equations; parametric equations, polar coordinates, sequences and series; power series; Taylor and Maclaurin series, elementary multivariate calculus. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

This course provides a mathematically rigorous introduction to statistics, based on calculus. Introductory probability theory is covered, including probability distributions and densities, random variables, the central limit theorem, and counting methods. Statistical inference is then covered, including estimation and confidence intervals, hypothesis tests. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

Students are introduced to applied calculus intended for students interested in the life sciences. Topics include: differentiation and antidifferentiation of common functions, general differentiation rules, curve sketching, limits at infinity, growth of functions, implicit differentiation, related rates, and optimization. Classes 3 hrs. and recitation 1.5 hrs. per week.

This course is a continuation of MATH 1250. Topics include: the integral; methods and applications of integration; differential equations; and an introduction to multivariable calculus (functions of several variables, partial derivatives). Classes 3 hrs. and recitation 1.5 hrs. per week.

This is a first course in linear algebra intended for students in Science or Engineering. Topics include: complex numbers, geometric vectors in three dimensions, equations of line and planes, systems of equations, Gaussian elimination, matrix algebra, vector spaces, linear transformations: definition and examples, null space and range, eigenvalues and eigenvectors, and orthogonality. Classes 3 hours plus recitation 1 1/2 hours a week.

First order differential equations: separable equations; exact equations; integral equations; integrating factors; linear differential equations; modelling electric circuits. Second order differential equations: homogeneous linear equations; constant coefficient equations; Euler-Cauchy equations; Wronskian; non-homogeneous equations; undetermined coefficients; variation of parameters; modelling forced oscillations and resonance modelling electric circuits phasor methods for particular solutions. Power series solutions. Legendre’s equation. Laplace transform, inverse transform. Linearity; transforms of derivatives and integral; s-shifting; t-shifting; unit step. Differentiation and integration of Laplace Transforms. Partial fractions method for inverse Laplace Transform. Applications to systems of differential equations, convolutions, the delta function, impulse response, transfer function. Periodic driving functions and Laplace Transforms. Fourier series; even and odd functions; half range expressions; Complex Fourier series; applications to systems driven by various periodic functions (e.g., square, wave, saw tooth, etc.). The line spectrum. Classes 3 hours plus recitation 1 1/2 hours a week.

This course provides an overview of a number of topics in discrete mathematics including sets, set operations, basic number theory, modular arithmetic, logic, proof techniques such as mathematical induction and proof by contradiction, elementary counting techniques, and a brief introduction to probability and networks. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

Students use various mathematical concepts to define tools and address problems of fundamental status in Computing Science. Topics include automata, formal languages, formal logic and computability. Other topics may be considered, such as information coding, complexity, knowledge modelling, and automated reasoning.

Students discuss errors in numerical analysis, theoretical and practical considerations of numerical methods for approximations of derivatives, systems of linear equations, systems of non-linear equations, and approximation of functions using polynomial and piece-wise polynomial interpolation. Classes 3 hrs. and recitation 1.5 hrs. per week.

This course is a rigorous study of the metric topology of the real line, sequences of real numbers, continuity of functions on the real line, and sequences of functions on real line. Compactness in higher-dimensional Euclidean spaces and the concept of a metric will also be discussed. Additional topics may include: elementary asymptotics; power series; uniform convergence and uniform continuity; and Riemann sums and integration. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

Topics include: limits and continuity of functions of several variables, partial derivatives, and the chain rule, directional derivatives and gradient vector, the total differential, tangent planes and normals to a surface, higher order partial derivatives, extrema of functions of two variables, Lagrange multipliers, double integrals, iterated integrals, double integrals in polar coordinates, applications of double integrals, the triple integral, triple integrals in cylindrical and spherical coordinates, applications of triple integrals vector fields, divergence and curl of vector fields, line integrals, path-independent line integrals. Green’s theorem, Stokes’ theorem, and the divergence theorem. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

This course covers the probability theory which underlies fundamental statistical concepts. It assumes a good knowledge of first-year calculus, and may cover the following topics: probability, conditional probability, Bayes’ Theorem, random variables, order statistics, discrete distributions, continuous distributions, expected values, moments, and special distributions including the Poisson, normal, binomial, exponential, and gamma distributions. Classes 3 hrs. plus recitation 1 1/2 hrs. a week.

This course continues MATH 2301 with further concepts and theory of linear algebra. Topics include inner product spaces, orthogonality, Gram-Schmidt Process, linear transformations and their matrix representation, change of basis and similarity, further study of eigenvalues and eigenvectors, canonical forms, with applications to linear differential equations and quadratic forms. Classes 3 hrs. plus recitation 1.5 hrs. a week.

This is a continuation of CSCI 2308 [MATH 2308]. Students engage with advanced topics in numerical methods for approximations of derivatives, systems of linear equations, systems of non-linear equations, and approximation of functions using polynomial and piece-wise polynomial interpolation, numerical integration, and approximation of functions by linear least squares. Classes 3 hrs. and recitation 1.5 hrs. per week.

Theory of systems of linear differential equations, linear systems with constant coefficients, solution by matrix methods, applications. Nonlinear differential equations: existence and uniqueness of solutions, stability and the phase plane, Liapunov Method. Various equations occurring in applications are qualitatively analyzed, Chaos and bifurcation.

Students study mathematical foundations of statistics, including both parametric and non-parametric inferences. Emphasis is placed on the properties of random variables and their distributions. The estimation of parameters by using sample statistics and tests of related hypotheses are included. Applications to computer science are studied.

Many important ideas of modern mathematics, such as the axiomatic method, emerged from the study of geometry. Students examine topics in geometry from Euclid to the present day, which may include axiomatic geometry, constructive geometry, inversive geometry, projective geometry, non-Euclidean geometry, and combinatorial geometry.

Topics include; open and closed sets in metric spaces, boundedness, total boundedness, compactness, sequences, completeness, continuity, uniform continuity, sequences of functions, pointwise and uniform convergence, metric spaces of functions, theorems of Baire, Arzela-Ascoli, and Stone-Weierstrass.

Students consider the numerical solution of initial value ordinary differential equations. Topics may include multi-step methods, Runge-Kutta methods, stability, stiffness, step-size selection, local error, etc.

This course is concerned with the numerical solution of boundary value ordinary differential equations. Topics may include finite difference methods, shooting methods, collocation methods, conditioning, mesh selection, error estimation, etc.

The study of algebraic structures, such as groups, rings, fields, posets, graphs, or universal algebras. The major emphasis is on derivation of theory, with inclusion of applications and examples.

This course is a further study of algebraic structures and their applications.

Students are introduced to various enumeration techniques and will include such topics as permutations and combinations, recurrence relations and generating functions. Various finite structures and their applications are also studied.

Various graph theoretic algorithms and their application to different problems are discussed. Topics are chosen from the following: the connector problem, the shortest path problem, the Chinese Postman problem and Euler trails, matchings and their applications to the personnel and optimal

This course will begin with a study of the topology of ordering and ordinals, and indexed unions, intersections, and products. Topics will include bounded and totally bounded sets, completeness and fixed point theorems. Following this, abstract topological spaces will be studied.

The complex plane. Elementary transformations and mappings, analytic functions, infinite series and uniform convergence. Differentiation and integration in the complex plane, residue. Harmonic functions, entire and meromorphic functions. Some principles of conformal mapping theory.

A continuation of MATH 4436. Further study of analytic functions and conformal mapping theory.

This course includes further topics on metric spaces. Topics include: Baire category theorem, the space of continuous functions, fixed points and integral equations, Arzela-Accoli theorem, the Stone-Weierstrass theorem, Picard existence theorem for differential equations, Riemann Integrability, sets of measure zero, and Lebesgue Theorem.

Research project in the mathematical sciences carried out by the student under the supervision of any member of the Department. The student will submit a thesis and present it orally. This course is open to 4th year honours students. Directed study 6 hrs. a week. 2 semesters.