University of Guelph

1996-97 Undergraduate Calendar


XII--Course Descriptions


Department of Mathematics and Statistics

Suggested initial course sequence: 1. For students with no OAC Calculus; 63-100, 63-101. 2. For students interested in the Mathematical Sciences, Chemistry, Physics; 63-120, 63-121. 3. For students interested in applications to the biological sciences; 63-108, 63-208.

63-100 Introductory Calculus I F,W(3-1)

A brief introduction to analytic geometry. The differential and integral calculus for algebraic, logarithmic, exponential and trigonometric functions, with applications to the natural and social sciences.

Prerequisites: OAC or equivalent Mathematics.
Exclusions: 63-108, 63-120, 66-111.

63-101 Introductory Calculus II W(3-1)

Transcendental functions; integration procedures and applications; polar coordinates; parametric equations; indeterminate forms. Introduction to partial derivatives and their geometric interpretation.

Prerequisites: 1 of 63-100, 63-108, 63-120, 66-111.
Exclusions: 63-121, 63-208, 66-121..

63-105 Introduction to Mathematical Modelling W(3-1)

The application of non-calculus techniques in modelling "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.

Exclusions: 27-190.

63-108 Elements of Calculus I F,W(3-1)

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.

Prerequisites: OAC Calculus or equivalent course.
Exclusions: 63-100, 63-120, 66-111.

63-120 Calculus I F(3-1)

This is a theoretical course intended primarily for mathematics, engineering and science students. Topics to be included are: inequalities and absolute values; limits and continuity using rigorous definitions the derivative and various applications, Rolle's theorem and the mean-value theorem for derivatives; the differential and antidifferentiation; the definite integral and various applications, the mean-value theorem for integrals, the fundamental theorem of calculus; logarithmic, exponential and elementary trigonometric functions.

Prerequisites: OAC Calculus or equivalent.
Exclusions: 63-100, 63-108, 66-111.

63-121 Calculus II S,W(3-1)

Topics to be included are: trigonometric and hyperbolic functions; techniques of integration; polar co-ordinates; parametric equations; indeterminate forms and improper integrals; elementary geometry of surfaces; a brief introduction to partial derivatives and multiple integrals.

Prerequisites: 63-120.
Exclusions: 63-101, 63-208, 66-121.

63-200 Set Theory F(3-1)

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.

Prerequisites: a calculus course at the university level.

63-208 Elements of Calculus II F,W(3-1)

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.

Prerequisites: 63-108 or 63-120 or 66-111.
Exclusions: 63-101, 63-121, 66-121.

63-213 Numerical Methods S,W(3-2)

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. A computer will be utilized in solving problem assignments.

Prerequisites: 1 of 63-101, 63-121, 63-208, 66-121, and prior experience in computer programming.

63-215 Applied Matrix Algebra F,W(3-1)

Matrices and matrix operations, matrix inverses, determinants, linear equations. N-dimensional vectors: dot product, linear independence, basis and dimension. Rank of a matrix. Eigenvalues, eigenvectors and diagonalization. Applications, including least squares.

Recommended: OAC Algebra and Geometry.

Exclusions: 63-216

63-216 Linear Algebra I F(3-0)

Matrix Notation, Matrix Arithmetic, Matrix Inverse and Determinant, Linear Systems of Equations, and Gaussian Elimination. The Basis Theory of Vector Spaces and Linear Transformations. Matrix Representations of Linear Transformations, Change of Basis, Diagonalization. Inner Product Spaces, Quadratic Forms, and Linear Differential Operators.

Prerequisites: 63-120 or equivalent.
Exclusions: 63-215

63-217 Differential Equations I W(3-1)

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.:a

Prerequisites: 1 of 63-101, 63-121, 63-208, 66-121.
Exclusions: 63-227.

63-220 Advanced Calculus I F(3-0)

Infinite sequences and series, tests for convergence, Taylor's series. Planes and quadric surfaces. Partial differentiation, directional derivatives and gradients, maxima and minima problems, Line integrals. Multiple integrals and coordinate transformations.

Prerequisites: 1 of 63-10163-12163-20866-121.
Exclusions: 63-210.

63-221 Advanced Calculus II W(3-0)

Implicit differentiation and implicit function theorems for systems of equations. Mappings and coordinate transformations. Vector and scalar fields including the gradient, divergence, curl and directional derivative, and their physical interpretation. Multiple Integrals. Theorems of Green and Stokes. An introduction to the theory of Riemann integration.

Prerequisites: 63-220.

63-227 Applied Differential Equations F(3-1)

Solution of differential equations which arise from problems in engineering. Linear equations of first and higher order; systems of linear equations; Laplace transforms: introduction to nonlinear equations and partial differential equations.

Prerequisites: 05-150, 63-121.
Exclusions: 63-217.

63-310 Differential Equations II F(3-1)

First order Linear Systems and their general solution by Matrix methods. Introduction to Nonlinear Systems, Stability, Limited 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.

Prerequisites: (63-213 or 76-244), (63-215 or 63-217), 63-227.

63-313 Modern Algebra F(3-0)

Symmetric Groups; the Affine Group of the Read Line; introduction to group theory; groups, subgroups, normal subgroups, factor groups, fundamental homomorphism theorem. Introduction to ring theory; rings, subrings, ideals, quotient rings, polynomial rings, Euclidean algorithm, fundamental ring homomorphism theorem. Introduction to Finite fields with Applications.

Prerequisites: 63-200, 63-215 or 63-216).

63-316 Linear Algebra II W(3-0)

Complex vector spaces. Direct sum decompositions, Cayley-Hamilton theorem, spectral theorem for normal operators, Jordan canonical form of a matrix.

Prerequisites: 63-216.

63-317 Partial Differential Equations and Special Functions (formerly: ) W(3-0)

Wave equation, heat equation, Laplace equation, linearity and separation of variables; solution by Fourier series; cylindrical and spherical geometires; Bessel and Legendre functions; Fourier transforms.

Prerequisites: 63-200,63-310.

63-320 Real Analysis I F(3-0)

Metric spaces and Normed Linear Spaces. Fixed Point Theorems and Applications. Sequences and Series of Functions. Riemann-Stieltjes integration.

Prerequisites: 63-200, 63-21663-221

63-324 Operations Research F(3-0)

Mathematical models, Introduction to linear and non-linear programming, the simplex method, convexity, Kuhn-Tucker condition. Game theory, decision analysis, and network analysis. Gueuing theory, birth and death processes.

Prerequisites: (63-215 or 63-216), 63-320, one statistics course.

63-326 Complex Analysis W(3-0)

The complex derivative and planar mappings. Analytic and harmonic functions. Conformal mappings. Elementary functions. Conformal mappings. Elementary functions. Cauchy-Goursat theorem. The Taylor and Laurent series. Calculus of residues with emphasis on applications.

Prerequisites: 63-200, 63-220.

63-351 Biomathematics (formerly: ) W(3-0)

Development, analysis, and interpretation of mathematical models of biological phenomena. Emphasis will be on deterministic discrete and continuous models.

Prerequisites: (63-21563-216), (63-217 or 63-227), at least 1 statistics course credit at the 200-level or above.

63-400 Advanced Differential Equations F(3-0)

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. Limited cycles and Poincare-Bendixson Theorem. Introduction to bifurcations and chaotic dynamics.

Prerequisites: 63-310,63-316 or 63-320.

63-405 Topics in Mathematics I W(3-0)

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.)

Prerequisites: 63-216, 63-220.

63-406 Topics in Mathematics II W(3-0)

Discussion of selected topics at an advanced level as in 63-405, but with different choice of topic.

Prerequisites: 63-216, 63-220.

63-407 Case Studies in Modelling F(2-2)

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, nonlinear programming, and dynamical systems theory. 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 modelling project in conjunction with the departmental Mathematics and Statistics Clinic. For further information concerning the Clinic, consult the department. (Offered in even-numbered years.)

Prerequisites: 7 course credits in mathematical science including 63-213.

63-414 Applied Algebra W(3-0)

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.)

Prerequisites: 63-313.

63-420 Advanced Analysis F(3-0)

Stone-Weierstrass approximation theorem. compactness in function spaces. Introduction to complex dynamics and the Mandelbrot set. Multivariate differential calculus.

Prerequisites: 63-316,63-320, 63-326.

63-422 Applied Functional Analysis W(3-0)

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.)

Prerequisites: 63-216,63-320.

63-424 Advanced Topics in Modelling W(3-0)

A series of modules on advanced topics in mathematical modelling techniques. Emphasis will be placed on the efficient use of computer resources and appropriate software, and the preparation of a written and an oral report. The module topics may vary from year to year, but examples include nonlinear regression, neural networks, pattern recognition, fuzzy set theory, and optimal control theory. Application areas will include the physical and biological science, engineering and environmetrics.

Prerequisites: (63-310 or 63-351), 63-324, 2 statistics courses. (63-443 recommended)

63-427 Advanced Partial Differential Equations F(3-0)

Theory of 1st and 2nd order partial differential equations with examples. Theory and examples of associated boundary value problems.

Prerequisites: (63-210 or 63-220), (63-217 or 63-227).

63-429 Geometry and Topology W(3-0)

Classical geometry of the plane and 3-space. Non-Euclidean geometics. Elementary topology of graphs and surfaces. Topics to be selected from: Algebraic geometry; Analysis on manifolds; Reimannian geometry; Tensor analysis; Homotopy and homology groups. (Offered in odd-numbered years.)

Prerequisites: 63-216,63-313, 63-320.

63-443 Advanced Numerical Methods F(3-0)

Numerical solution of linear systems, differential equations; the algebraic eigenvalue problem, interpolation and approximation of functions, numerical quadrature.

Prerequisites: 63-213, (63-215 or 63-216), 63-220, (63-217 or 63-227).

63-451 Environmental Transport and Dynamics F(3-0)

Mathematical modelling 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.)

Prerequisites: 63-351 or 63-310, 1 statistics course.

1996-97 Undergraduate Calendar
XII--Course Descriptions

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Last revised: August 28, 1996. Contact: