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*1000, one STAT*XXXX course.
MATH*1000 Introductory Calculus F,W (30) [0.50] 
A brief introduction to analytical geometry. The differential and integral calculus for algebraic, logarithmic, exponential
and trigonometric functions, with applications. (Also offered through distance education format.)

Prerequisite(s): 
1 4U credit in mathematics or 1 OAC credit in mathematics 
Restriction(s): 
MATH*1080 or MATH*1200. Not available to students registered in the B.Sc. and B.Sc. (Agr.) programs.

MATH*1050 Introduction to Mathematical Modeling U (31) [0.50] 
The application of noncalculus techniques in modeling "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 majors and minors. 
MATH*1080 Elements of Calculus I F,W (31) [0.50] 
The elements of the calculus of one variable with illustration and emphasis on its application in the biological sciences.
The elementary functions, sequences and series, difference equations, differential and integral calculus.

Prerequisite(s): 
1 of 4U Advanced Functions , 4U Advanced Functions and Calculus or OAC Calculus 
Restriction(s): 
MATH*1000, 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 OAC Calculus 
Restriction(s): 
MATH*1000, MATH*1080 
MATH*1210 Calculus II W (31) [0.50] 
Topics include inverse functions, inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, indeterminate
forms and l'Hopital's rule; techniques of integration; parametric equations; polar coordinates; introduction to MacLaurin
and Taylor series; functions of several variables; and partial derivatives.

Prerequisite(s): 
1 of MATH*1000, MATH*1080, MATH*1200 
Restriction(s): 
MATH*2080 
MATH*2000 Set Theory F (31) [0.50] 
The algebra of sets. Equivalence relations, mappings and inverse mappings. Review of the real number system. Countable and
uncountable sets. Partially and totally ordered sets. Complex numbers and their arithmetic. Geometry and topology of the line
and the plane. Emphasis is placed on developing skills in constructing mathematical proofs.

Prerequisite(s): 
0.50 credits in calculus at the university level 
MATH*2080 Elements of Calculus II F,W (31) [0.50] 
Techniques of integration, introduction to differential equations and the elements of multivariate calculus. Illustrations
and emphasis will be on biological applications. An introduction to vectors, multivariable and vector functions, difference
equations, partial differentiation and multiple integration.

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 an overview of and practical experience in utilizing algorithms for solving numerical problems arising
in applied sciences. Topics covered will include solution of a single nonlinear equation, interpolation, numerical differentiation
and integration, solution of differential equations and systems of linear algebraic equations. Students will utilize computers
in solving problem assignments.

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. 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*2200 Advanced Calculus I F (30) [0.50] 
Infinite sequences and series of numbers, power series, tests for convergence; Taylor's theorem and Taylor series for functions
of one variable; planes and quadratric surfaces; limits, continuity, and differentiability; partial differentiation, directional
derivatives and gradients; tangent planes, linear approximation, and Taylor's theorem for functions of two variables; critical
points, extreme value problems; implicit function theorem; Jacobians; double integrals, iterated integrals and change of variables.

Prerequisite(s): 
MATH*1210 or MATH*2080 
MATH*2210 Advanced Calculus II W (30) [0.50] 
Spherical and cylindrical polar coordinate transformations; multiple integrals; line integrals; vector and scalar fields including
the gradient, divergence, curl and directional derivative, and their physical interpretation; theorems of Green and Stokes;
uniform convergence.

Prerequisite(s): 
MATH*2200, (MATH*1200 is strongly recommended)

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] 
Symmetric groups; introduction to group theory; groups, subgroups, normal subgroups, factor groups, fundamental homomorphism
theorem. Introduction to ring theory; rings, subrings, ideals, quotient rings, polynomial rings, fundamental ring homomorphism
theorem.

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

MATH*3240 Operations Research F (30) [0.50] 
Mathematical models. Linear programming and sensitivity analysis. Network analysis: shortest path, maximum flow and minimal
spanning tree problems. Introduction to nonlinear programming. Constrained optimization: the FrankWolfe method. Deterministic
and probabilistic dynamic programming.

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

Corequisite(s): 
MATH*2200 
MATH*3260 Complex Analysis W (30) [0.50] 
The complex derivative and planar mappings. Analytic and harmonic functions. Conformal mappings. Elementary functions. CauchyGoursat
theorem. The Taylor and Laurent series. Calculus of residues with emphasis on applications.

Prerequisite(s): 
MATH*2000, MATH*2200 
MATH*4000 Advanced Differential Equations W (30) [0.50] 
A rigorous treatment of the qualitative theory of ordinary differential equations and an introduction to the modern theory
of dynamical systems, existence, uniqueness, and continuity theorems. Definition and properties of dynamical systems. Linearization
and local behaviour of nonlinear systems. Stable Manifold theorem. Liapunov stability. Limit cycles and PoincaréBendixson
Theorem. 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] 
Study of selected topics in applied mathematics at an advanced level, intended mainly for mathematical science students in
the 7th or 8th semester. Sample topics are optimal control theory and nonlinear programming. The course will include case
studies of realworld problems arising from various areas and the contribution of mathematical models to their solution. Part
of the course requirement will involve the completion of a mathematical modeling project in conjunction with the departmental
Mathematics and Statistics Clinic. For further information concerning the Clinic, consult the department. (Offered in evennumbered
years.)

Prerequisite(s): 
3.50 credits in mathematical science including MATH*2130 
MATH*4140 Applied Algebra W (30) [0.50] 
Finite symmetric groups, dihedral and cyclic groups with applications to the group of symmetries of a geometric figure in
the plane. PolyaBurnside method of enumeration with applications. Galois fields with applications to combinatorial design
constructions. Error correcting binary codes. (Offered in evennumbered years.)

Prerequisite(s): 
MATH*3130 
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] 
A study of selected advanced topics in mathematical modeling, to include model formulation, techniques of model analysis and
interpretation of results. Topics usually include transportation and assignment problems, minimum cost flow problems and network
simplex methods, Markov chains, queuing theory. Student participation in researching a project and in the preparation of a
report. (Offered in oddnumbered years.)

Prerequisite(s): 
MATH*3240 
MATH*4290 Geometry and Topology W (30) [0.50] 
Classical geometry of the plane and 3space. NonEuclidean geometries. Elementary topology of graphs and surfaces. Topics
to be selected from: algebraic geometry; analysis on manifolds; Riemannian geometry; tensor analysis; homotopy and homology
groups. (Offered in oddnumbered years.)

Prerequisite(s): 
MATH*3130, MATH*3200 
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) [0.50] 
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. 