# XII. Course Descriptions

## Mathematics

Department of Mathematics and Statistics

Suggested initial course sequence:

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

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

3. For students not expecting to pursue further studies in mathematics: MATH*1000, one STAT*XXXX course

MATH*1000 Introductory Calculus F,W (3-0) [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*1080 Elements of Calculus I F,W (3-1) [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): 4U Advanced Functions and Calculus or OAC Calculus
Restriction(s): MATH*1000, MATH*1200, IPS*1110
MATH*1200 Calculus I F (3-1) [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; compund 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 mean-value 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): 4U Advanced Functions and Calculus or OAC Calculus
Restriction(s): MATH*1000, MATH*1080, IPS*1110
MATH*1210 Calculus II W (3-1) [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. (Last offering Summer 2008)
Prerequisite(s): 1 of MATH*1000, MATH*1080, MATH*1200
Restriction(s): MATH*2080
MATH*2000 Set Theory F (3-1) [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 (3-1) [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 S,W (3-2) [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 (3-1) [0.50]
Matrices and matrix operations, matrix inverse and determinant, linear equations. N-dimensional vectors: dot product, linear independence, basis and dimension. Rank of a matrix. Eigenvalues, eigenvectors and diagonalization. Applications, including least squares. (Also offered through distance education format.)
Prerequisite(s): 1 of a 4U mathematics credit, an OAC mathematics credit, first year university mathematics credit
Restriction(s): MATH*2160
MATH*2160 Linear Algebra I F (3-0) [0.50]
Matrix notation, matrix arithmetic, matrix inverse and determinant, linear systems of equations, and Gaussian elimination. The basic theory of vector spaces and linear transformations. Matrix representations of linear transformations, change of basis, diagonalization. Inner product spaces, quadratic forms, orthogonalization and projections.
Prerequisite(s): (IPS*1110 or MATH*1200), (1 of MATH*2150, 4U Geometry and Discrete Mathematics, OAC Algebra and Geometry)
MATH*2170 Differential Equations I W,S (3-1) [0.50]
First order equations, linear equations of second and higher orders, phase plane, difference equations, introduction to power series methods, Laplace transforms, formulation, solution and interpretation of differential equations of interest in science.
Prerequisite(s): MATH*1210 or MATH*2080
Restriction(s): MATH*2270
MATH*2200 Advanced Calculus I F (3-0) [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): 1 of IPS*1210, MATH*1210, MATH*2080
MATH*2210 Advanced Calculus II W (3-0) [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*2270 Applied Differential Equations F (3-1) [0.50]
Solution of differential equations which arise from problems in engineering. Linear equations of first and higher order; systems of linear equations; Laplace transforms; series solutions of second-order equations; introduction to partial differential equations.
Prerequisite(s): ENGG*1500, MATH*1210
Restriction(s): MATH*2170
MATH*3100 Differential Equations II F (3-1) [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 (3-0) [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*3160 Linear Algebra II W (3-0) [0.50]
Complex vector spaces. Direct sum decompositions, Cayley-Hamilton theorem, spectral theorem for normal operators, Jordan canonical form of a matrix.
Prerequisite(s): MATH*2160
MATH*3170 Partial Differential Equations and Special Functions W (3-0) [0.50]
Wave equation, heat equation, Laplace equation, linearity and separation of variables; solution by Fourier series; Bessel and Legendre functions; Fourier transforms; introduction to the method of characteristics.
Prerequisite(s): MATH*2000, MATH*3100
MATH*3200 Real Analysis F (3-0) [0.50]
Metric spaces and normed linear spaces. Fixed point theorems with applications to fractals. Uniform continuity. Riemann-Stieltjes integration.
Prerequisite(s): MATH*2000, MATH*2160, MATH*2210
MATH*3240 Operations Research F (3-0) [0.50]
Mathematical models. Linear programming and sensitivity analysis. Network analysis: shortest path, maximum flow and minimal spanning tree problems. Introduction to non-linear programming. Constrained optimization: the Frank-Wolfe method. Deterministic and probabilistic dynamic programming.
Prerequisite(s): (MATH*2150 or MATH*2160), 0.50 credits in statistics
Co-requisite(s): MATH*2200
MATH*3260 Complex Analysis W (3-0) [0.50]
The complex derivative and planar mappings. Analytic and harmonic functions. Conformal mappings. Elementary functions. Cauchy-Goursat theorem. The Taylor and Laurent series. Calculus of residues with emphasis on applications.
Prerequisite(s): MATH*2000, MATH*2200
MATH*3510 Biomathematics W (3-0) [0.50]
Development, analysis, and interpretation of mathematical models of biological phenomena. Emphasis will be on deterministic discrete and continuous models.
Prerequisite(s): (MATH*2150 or MATH*2160), (MATH*2170 or MATH*2270), at least 0.50 credits in statistics at the 2000 level or above
MATH*4000 Advanced Differential Equations W (3-0) [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 Poincaré-Bendixson Theorem. Introduction to bifurcations and chaotic dynamics. (Offered in even-numbered years.)
Prerequisite(s): MATH*3100, (MATH*3160 or MATH*3200)
MATH*4050 Topics in Mathematics I W (3-0) [0.50]
Discussion of selected topics at an advanced level. Intended mainly for mathematics students in the 6th to 8th semester. Content will vary from year to year. Sample topics: probability theory, Fourier analysis, mathematical logic, operator algebras, number theory combinatorics, philosophy of mathematics, fractal geometry, chaos, stochastic differential equations. (Offered in odd-numbered years.)
Prerequisite(s): MATH*2160, MATH*3200
MATH*4060 Topics in Mathematics II W (3-0) [0.50]
Discussion of selected topics at an advanced level as in MATH*4050, but with different choice of topic. (Offered in even-numbered years.)
Prerequisite(s): MATH*2160, MATH*3200
MATH*4070 Case Studies in Modeling F (2-2) [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 real-world 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 even-numbered years.)
Prerequisite(s): 3.50 credits in mathematical science including MATH*2130
MATH*4140 Applied Algebra W (3-0) [0.50]
Finite symmetric groups, dihedral and cyclic groups with applications to the group of symmetries of a geometric figure in the plane. Polya-Burnside method of enumeration with applications. Galois fields with applications to combinatorial design constructions. Error correcting binary codes. (Offered in even-numbered years.)
Prerequisite(s): MATH*3130
MATH*4200 Advanced Analysis F (3-0) [0.50]
Sequences and series of functions. Stone-Weierstrass approximation theorem. Compactness in function spaces. Introduction to complex dynamics and the Mandelbrot set. Multivariate differential calculus. (Offered in even-numbered years.)
Prerequisite(s): MATH*3160, MATH*3200, MATH*3260
MATH*4220 Applied Functional Analysis W (3-0) [0.50]
Hilbert and Banach spaces: applications to Fourier series and numerical analysis. Hahn-Banach theorem; weak topologies. Generalized functions; application to differential equations. Completeness; uniform boundedness principle. Lebesque measure and integral; applications to probability and dynamics. Spectral theory. (Offered in even-numbered years.)
Prerequisite(s): MATH*2160, MATH*3200
MATH*4240 Advanced Topics in Modeling W (3-0) [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 odd-numbered years.)
Prerequisite(s): MATH*3240
MATH*4270 Advanced Partial Differential Equations F (3-0) [0.50]
Theory of 1st and 2nd order partial differential equations with examples. Classification of linear second order PDE. Theory and examples of associated boundary value problems. Maximum principles. Green's functions. Introduction to nonlinear PDE. Applications.
Prerequisite(s): MATH*3170, MATH*3200, MATH*3260
MATH*4290 Geometry and Topology W (3-0) [0.50]
Classical geometry of the plane and 3-space. Non-Euclidean 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 odd-numbered years.)
Prerequisite(s): MATH*3130, MATH*3200
MATH*4430 Advanced Numerical Methods F (3-0) [0.50]
Numerical solution of linear systems, differential equations; the algebraic eigenvalue problem, interpolation and approximation of functions, numerical quadrature. (Offered in odd-numbered years.)
Prerequisite(s): MATH*2130, (MATH*2150 or MATH*2160), MATH*2200, (MATH*2170 or MATH*2270)
MATH*4510 Environmental Transport and Dynamics F (3-0) [0.50]
Mathematical modeling of environmental transport systems. Linear and nonlinear compartmental models. Convective and diffusive transport. Specific models selected from hydrology; ground-water 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 odd-numbered years.)
Prerequisite(s): MATH*3510 or MATH*3100, 0.50 credits in statistics