XII. Course Descriptions
Mathematics
Department of Mathematics and Statistics
Suggested initial course sequence:

For students with 4U or OAC Calculus and expecting to pursue further studies in mathematics or the physical sciences: MATH*1200, MATH*1210.

For students interested in applications to the biological sciences: MATH*1080, MATH*2080.

For students not expecting to pursue further studies in mathematics: MATH*1030, one STAT*XXXX course.
MATH*1030 Business Mathematics F,W (31) [0.50] 
This course is intended for business and economics students. The topics covered include lines, systems of linear equations,
convex sets, and basic algebra including exponential and logarithmic functions. Calculus covered in the course includes limits,
continuity, sequences and series, derivatives, higher order derivatives, curve sketching, linear approximators, optimization,
and integration. (Also offered through Distance Education format.)

Prerequisite(s): 
4U Advanced Functions 
Restriction(s): 
MATH*1000, MATH*1080, MATH*1200. Not available to students registered in the B.Sc. program.

MATH*1050 Introduction to Mathematical Modeling U (31) [0.50] 
This course applies noncalculus techniques to model "real world" problems in business, psychology, sociology, political science
and ecology. The mathematical topics introduced include graphs and directed graphs, linear programming, matrices, probability,
games and decisions, and difference equations. Mathematics majors may not take this course for credit.

Equate(s): 
CIS*1900 
Restriction(s): 
Not available to students registered in B.Comp programs or CIS. 
MATH*1080 Elements of Calculus I F,W (31) [0.50] 
This course provides an introduction to the calculus of one variable with emphasis on mathematical modelling in the biological
sciences. The topics covered include elementary functions, sequences and series, difference equations, differential calculus
and integral calculus.

Prerequisite(s): 
1 of 4U Advanced Functions, 4U Advanced Functions and Calculus or equivalent 
Restriction(s): 
MATH*1000, MATH*1030, MATH*1200 
MATH*1200 Calculus I F (31) [0.50] 
This is a theoretical course intended primarily for students who expect to pursue further studies in mathematics and its applications.
Topics include inequalities and absolute value; compound angle formulas for trigonometric functions; limits and continuity
using rigorous definitions; the derivative and derivative formulas (including derivatives of trigonometric, exponential and
logarithmic functions); Fermat's theorem; Rolle's theorem; the meanvalue theorem; applications of the derivative; Riemann
sums; the definite integral; the fundamental theorem of calculus; applications of the definite integral; the mean value theorem
for integrals.

Prerequisite(s): 
1 of 4U Calculus and Vectors, 4U Advanced Functions and Calculus or Grade 12 Calculus 
Restriction(s): 
MATH*1000, MATH*1080 
MATH*1210 Calculus II W (31) [0.50] 
This course is a continuation of MATH*1200. It is a theoretical course intended primarily for students who need or expect to pursue further studies in mathematics,
physics, chemistry, engineering and computer science. Topics include inverse functions, inverse trigonometric functions, hyperbolic
functions, indeterminate forms and l'Hopital's rule, techniques of integration, parametric equations, polar coordinates, Taylor
and Maclaurin series; functions of two or more variables, partial derivatives, and if time permits, an introduction to multiple
integration.

Prerequisite(s): 
1 of MATH*1000, MATH*1080, MATH*1200 
Restriction(s): 
MATH*2080 
MATH*2000 Set Theory F (31) [0.50] 
This course introduces the theory of sets and emphasizes formal mathematical proof. Topics include relations and functions,
number systems including formal properties of the natural numbers, integers, and the real and complex numbers. Equivalence
relations and partial and total orders are introduced. The geometry and topology of the real number line and Cartesian plane
are introduced. Techniques of formal proof are introduced including wellordering, mathematical induction, proof by contradiction,
and proof by construction.

Prerequisite(s): 
0.50 credits in mathematics at the university level 
MATH*2080 Elements of Calculus II F,W (31) [0.50] 
This course will expand on integration techniques, and introduce students to difference and differential equations, vectors,
vector functions, and elements of calculus of two or more variables such as partial differentiation and multiple integration.
The course will emphasize content relevant to analyzing biological systems, and methods will be illustrated by application
to biological systems.

Prerequisite(s): 
1 of MATH*1000, MATH*1080, MATH*1200 
Restriction(s): 
MATH*1210 
MATH*2130 Numerical Methods W (32) [0.50] 
This course provides a theoretical and practical introduction to numerical methods for approximating the solution(s) of linear
and nonlinear problems in the applied sciences. The topics covered include: solution of a single nonlinear equation; polynomial
interpolation; numerical differentiation and integration; solution of initial value and boundary value problems; and the solution
of systems of linear and nonlinear algebraic equations.

Prerequisite(s): 
MATH*1210 or MATH*2080 
MATH*2150 Applied Matrix Algebra S,F,W (31) [0.50] 
This course provides an introduction to linear algebra in Euclidean space. Topics covered include: Ndimensional vectors,
dot product, matrices and matrix operations, systems of linear equations and Gaussian elimination, linear independence, subspaces,
basis and dimension, matrix inverse, matrix rank and determinant, eigenvalues, eigenvectors and diagonalization. Applications
of these topics, including least squares fitting, will be included. (Also offered through Distance Education format.) MATH*2150
is not intended for Mathematics majors.

Prerequisite(s): 
1 of a 4U mathematics credit or a first year university mathematics credit 
Restriction(s): 
MATH*2160 
MATH*2160 Linear Algebra I F (30) [0.50] 
This course provides an introduction to linear algebra and vector spaces and emphasizes formal mathematical proof. Topics
covered include: Ndimensional vectors, inner products, matrices and matrix operations, systems of linear equations and Gaussian
elimination, the basic theory of vector spaces and linear transformations, matrix representations of linear transformations,
change of basis matrices, eigenvalues, eigenvectors and diagonalization, inner product spaces, quadratic forms, orthogonalization
and projections.

Prerequisite(s): 
MATH*1200 
Restriction(s): 
MATH*2150 
MATH*2170 Differential Equations I W,S (31) [0.50] 
This course introduces the theory and application of differential equations, which are used to describe phenomena in a wide
range of areas. First order equations are studied extensively as well as linear equations of second and higher order. Other
topics include difference equations, phase plane analysis, and an introduction to power series methods and Laplace transforms.

Prerequisite(s): 
MATH*1210 or MATH*2080 
Restriction(s): 
MATH*2270 
MATH*2200 Advanced Calculus I F (30) [0.50] 
The topics covered in this course include infinite sequences and series, power series, tests for convergence, Taylor's theorem
and Taylor series for functions of one variable, planes and quadratric surfaces, limits, and continuity, differentiability
of functions of two or more variables, partial differentiation, directional derivatives and gradients, tangent planes, linear
approximation, Taylor's theorem for functions of two variables, critical points, extreme value problems, implicit function
theorem, Jacobians, multiple integrals, and change of variables.

Prerequisite(s): 
MATH*1210 or MATH*2080 
MATH*2210 Advanced Calculus II W (30) [0.50] 
This course continues the study of multiple integrals, introducing spherical and cylindrical polar coordinates. The course
also covers vector and scalar fields, including the gradient, divergence, curl and directional derivative, and their physical
interpretation, as well as line integrals and the theorems of Green and Stokes.

Prerequisite(s): 
MATH*2200 
MATH*2270 Applied Differential Equations F (31) [0.50] 
This course covers the solution of differential equations which arise from problems in engineering. Topics include linear
equations of first and higher order, systems of linear equations, Laplace transforms, series solutions of secondorder equations,
and an introduction to partial differential equations. This course is intended for students in B.Eng.

Prerequisite(s): 
ENGG*1500, MATH*1210 
Restriction(s): 
MATH*2170 
MATH*3100 Differential Equations II F (31) [0.50] 
First order linear systems and their general solution by matrix methods. Introduction to nonlinear systems, stability, limit
cycles and chaos using numerical examples. Solution in power series of second order equations including Bessel's equation.
Introduction to partial differential equations and applications.

Prerequisite(s): 
(MATH*2150 or MATH*2160), MATH*2170 
MATH*3130 Abstract Algebra F (30) [0.50] 
This course is an introduction to abstract algebra, covering both group theory and ring theory. Specific topics covered include
an introduction to group theory, permutations, symmetric and dihedral groups, subgroups, normal subgroups and factor groups.
Group theory continues through the fundamental homomorphism theorem. Ring theory material covered includes an introduction
to ring theory, subrings, ideals, quotient rings, polynomial rings, and the fundamental ring homomorphism theorem.

Prerequisite(s): 
MATH*2000, (MATH*2150 or MATH*2160)

MATH*3160 Linear Algebra II W (30) [0.50] 
The topics in this course include complex vector spaces, direct sum decompositions of vector spaces, the CayleyHamilton theorem,
the spectral theorem for normal operators and the Jordan canonical form.

Prerequisite(s): 
MATH*2160 
MATH*3200 Real Analysis F (30) [0.50] 
This course provides a basic foundation for real analysis. The rigorous treatment of the subject in terms of theory and examples
gives students the flavour of mathematical reasoning and intuition for other advanced topics in mathematics. Topics covered
include the real number line and the supremum property; metric spaces; continuity and uniform continuity; completeness and
compactness; the Banach fixedpoint theorem and its applications to ODEs; uniform convergence and the rigorous treatment of
the Riemann integral.

Prerequisite(s): 
MATH*2000, MATH*2160, MATH*2210 
MATH*3240 Operations Research F (30) [0.50] 
This is a course in mathematical modelling which has applications to engineering, economics, business and logistics. Topics
covered include linear programming and the simplex method, network models and the shortest path, maximum flow and minimal
spanning tree problems as well as a selection of the following: nonlinear programming, constrained optimization, deterministic
and probabilistic dynamic programming, game theory and simulation.

Prerequisite(s): 
(MATH*2150 or MATH*2160), 0.50 credits in statistics

Corequisite(s): 
MATH*2200 
MATH*3260 Complex Analysis W (30) [0.50] 
This course extends calculus to cover functions of a complex variable; it introduces complex variable techniques which are
very useful for mathematics, the physical sciences and engineering. Topics include complex differentiation, planar mappings,
analytic and harmonic functions, contour integration, Taylor and Laurent series, the residue calculus and its application
to the computation of trigonometric and improper integrals, conformal mapping and the Dirichlet problem.

Prerequisite(s): 
MATH*2200 
MATH*3510 Biomathematics W (30) [0.50] 
This course will convey the fundamentals of applying mathematical modelling techniques to understanding and predicting the
dynamics of biological systems. Students will learn the development, analysis, and interpretation of biomathematical models
based on discretetime and continuoustime models. Applications may include examples from population biology, ecology, infectious
diseases, microbiology, and genetics.

Prerequisite(s): 
(MATH*2150 or MATH*2160), (MATH*2170 or MATH*2270)

MATH*4000 Advanced Differential Equations W (30) [0.50] 
This course provides a rigorous treatment of the qualitative theory of ordinary differential equations and an introduction
to the modern theory of dynamical systems. Existence and uniqueness of solutions and their dependence on initial conditions
and parameters are covered as well as linearization and the local behaviour of nonlinear systems near equilibrium points.
Stability of solutions is examined including the stable manifold theorem and the method of Lyapunov. Limit cycles are covered,
with a discussion of PoincaréBendixson theory in the plane. The definition and a discussion of some properties of dynamical
systems, both continuous and discrete, are given, including an introduction to bifurcations and chaotic dynamics. (Offered
in evennumbered years.)

Prerequisite(s): 
MATH*3100, (MATH*3160 or MATH*3200)

MATH*4050 Topics in Mathematics I W (30) [0.50] 
In this course students will discuss selected topics at an advanced level. It is intended mainly for mathematics students
in the 6th to 8th semester. Content will vary from year to year. Sample topics include: probability theory, Fourier analysis,
mathematical logic, operator algebras, number theory combinatorics, philosophy of mathematics, fractal geometry, chaos, stochastic
differential equations. (Offered in oddnumbered years.)

Prerequisite(s): 
MATH*3200 
MATH*4070 Case Studies in Modeling F (22) [0.50] 
The course covers selected case studies in mathematical modelling at an advanced level, and is intended for mathematical science
students in the 7th or 8th semester. The course covers case studies of realworld problems arising from various areas and
the contribution of mathematical models to their solution. Examples that may be covered include models of data communication
networks, transportation networks, and spread of epidemics. (Offered in evennumbered years.)

Prerequisite(s): 
3.50 credits in mathematical science including MATH*2130 and (MATH*2170 or MATH*2270)

MATH*4140 Applied Algebra W (30) [0.50] 
The topics covered in this course include permutation representations and the PolyaBurside technique of enumeration, classification
of groups, the theory of fields including Galois theory, the construction of finite fields, combinatorial applications to
the design of experiments, the theory of binary error correcting codes, combinatorial graphs and their symmetry groups, and
finite combinatorial geometries. (Offered in evennumbered years.)

Prerequisite(s): 
MATH*3130 
MATH*4200 Advanced Analysis F (30) [0.50] 
This course covers advanced topics in analysis. It includes Lebesgue measure and integration, measuretheoretic probability,
sequences and series of functions, the StoneWeierstrass approximation theorem, compactness in function spaces and the implicit
and inverse function theorems. (Offered in evennumbered years.)

Prerequisite(s): 
MATH*3160, MATH*3200 
MATH*4220 Applied Functional Analysis W (30) [0.50] 
Hilbert and Banach spaces are covered including applications to Fourier series and numerical analysis. Other topics include
the HahnBanach theorem; weak topologies; generalized functions and their application to differential equations; completeness;
the uniform boundedness principle; Lebesgue measure and integral and applications to probability and dynamics; and spectral
theory. (Offered in evennumbered years.)

Prerequisite(s): 
MATH*3200 
MATH*4240 Advanced Topics in Modeling W (30) [0.50] 
This course presents selected advanced topics in mathematical modelling, such as model formulation, techniques of model analysis
and interpretation of results. Topics may include discrete and continuous models, both deterministic and probabilistic. (Offered
in oddnumbered years.)

Prerequisite(s): 
MATH*3240 
MATH*4270 Advanced Partial Differential Equations F (30) [0.50] 
This course focuses on the theory of firstorder and secondorder partial differential equations, with examples and applications
from selected fields such as physics, engineering and biology. It covers classification of linear secondorder partial differential
equations, the theory of associated boundary value problems, maximum principles and Green’s functions. It also introduces
nonlinear partial differential equations.

Prerequisite(s): 
MATH*3170, MATH*3200, MATH*3260 
MATH*4290 Geometry and Topology W (30) [0.50] 
This course introduces modern topics in geometry. Topics include the classical geometry of the plane and 3space, nonEuclidean
geometries, the elementary topology of graphs and surfaces and a selection from pointset topology, differential geometry,
algebraic geometry, analysis on manifolds, Riemannian geometry, tensor analysis, homotopy and homology groups. (Offered in
oddnumbered years.)

Prerequisite(s): 
MATH*3130, MATH*3200 
MATH*4430 Advanced Numerical Methods F (30) [0.50] 
This course covers a wide range of numerical methods for finding solutions to mathematical problems. A large component of
the course will be the implementation of algorithms on a computer using appropriate software. The mathematical problems addressed
include the solution of linear systems of equations via both direct and indirect methods, finding zeros of a nonlinear function,
the solution of ordinary differential equations, and the approximation of eigenvalues. Other topics may include numerical
quadrature, numerical differentiation, interpolation and approximation of functions, fast Fourier transforms, finite difference
and shooting methods for boundary value problems, and an introduction to partial differential equations. (Offered in oddnumbered
years.)

Prerequisite(s): 
MATH*2130, (MATH*2150 or MATH*2160), MATH*2200, (MATH*2170 or MATH*2270)

MATH*4510 Environmental Transport and Dynamics F (30) [0.50] 
Mathematical modeling of environmental transport systems. Linear and nonlinear compartmental models. Convective and diffusive
transport. Specific models selected from hydrology; groundwater and aquifer transport, dispersion of marine pollution, effluents
in river systems; atmospheric pollen dispersion, plume models, dry matter suspension and deposition; Global circulation: tritium
distribution. (Offered in oddnumbered years.)

Prerequisite(s): 
MATH*3510 or MATH*3100, 0.50 credits in statistics

MATH*4600 Advanced Research Project in Mathematics F,W (06) [1.00] 
Each student in this course will undertake an individual research project in some area of mathematics, under the supervision
of a faculty member. A written report and a public presentation of the project will be required.

Restriction(s): 
Approval of a supervisor and the course coordinator. 