University of Guelph 1999-2000 Undergraduate Calendar

XII--Course Descriptions


Mathematics

Department of Mathematics and Statistics.

Suggested initial course sequence:

  1. For students not expecting to pursue further studies in mathematics: MATH*1000, MATH*1010.
  2. For students with OAC Calculus and expecting to pursue further studies in mathematics: MATH*1200, MATH*1210.
  3. For students interested in applications to the biological sciences; MATH*1080, MATH*2080
  4. Students with credit in MATH*1200 may not proceed to MATH*1000; students with credit in MATH*2160 may not proceed to MATH*2150.

MATH*1000 Introductory Calculus I F,W(3-1) [0.50]

This course may be taken by students without credit in OAC Calculus. A brief introduction to analytic geometry. The differential and integral calculus for algebraic, logarithmic, exponential and trigonometric functions, with applications.

Prerequisites: 1 OAC credit in mathematics (grade 12 advanced level mathematics is strongly recommended)
Exclusions: MATH*1080, MATH*1200, IPS*1110

MATH*1010 Introductory Calculus II W(3-1) [0.50]

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

Prerequisites: 1 of MATH*1000, MATH*1080, MATH*1200, IPS*1110
Exclusions: MATH*1210, MATH*2080, IPS*1210

MATH*1050 Introduction to Mathematical Modelling W(3-1) [0.50]

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: CIS*1900

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.

Prerequisites: OAC Calculus or equivalent course
Exclusions: 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 need or expect to pursue further studies in mathematics and its applications. 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 MATH*1000 or equivalent
Exclusions: MATH*1080, IPS*1110

MATH*1210 Calculus II S,W(3-1) [0.50]

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.

Prerequisites: 1 of MATH*1000, MATH*1080, MATH*1200*, IPS*1110* (*preferred)
Exclusions: MATH*1010, MATH*2080, IPS*1210

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.

Prerequisites: 0.50 credit 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.

Prerequisites: 1 of MATH*1000, MATH*1080, MATH*1200, IPS*1110
Exclusions: MATH*1010, MATH*1210, IPS*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. A computer will be utilized in solving problem assignments.

Prerequisites: 1 of MATH*1010, MATH*1210, MATH*2080, IPS*1210 and prior experience in computer programming

MATH*2150 Applied Matrix Algebra F,W(3-1) [0.50]

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: 1 OAC mathematics credit or first year university mathematics credit.

Exclusions: 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 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.

Prerequisites: MATH*1200, (MATH*2150 or 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 pane, difference equations, introduction to power series methods, Laplace transforms, formulation, solution and interpretation of differential equations of interest in science.

Prerequisites: 1 of MATH*1010, MATH*1210, MATH*2080, IPS*1210
Exclusions: MATH*2270

MATH*2200 Advanced Calculus I F(3-0) [0.50]

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

Prerequisites: 1 of MATH*1010, MATH*1210, MATH*2080, IPS*1210
Exclusions: MATH*2100

MATH*2210 Advanced Calculus II W(3-0) [0.50]

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: 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; introduction to nonlinear equations and partial differential equations.

Prerequisites: ENGG*1500, MATH*1210
Exclusions: 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.

Prerequisites: (MATH*2130 or PHYS*2440), (MATH*2150 or MATH*2160), MATH*2170

MATH*3130 Algebraic Structures 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.

Prerequisites: 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.

Prerequisites: 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.

Prerequisites: MATH*2000, MATH*3100
Exclusion: MATH*3110

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.

Prerequisites: 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.

Prerequisites: (MATH*2150 or MATH*2160), MATH*2200, 0.50 credit in statistics
Exclusion: MATH*3420

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.

Prerequisites: 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.

Prerequisites: (MATH*2150 or MATH*2160), (MATH*2170 or MATH*2270), at least 0.50 credit in statistics at the 2000 level or above
Exclusion: MATH*3480

MATH*4000 Advanced Differential Equations F(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 Poincar-Bendixson Theorem. Introduction to bifurcations and chaotic dynamics.

Prerequisites: 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.)

Prerequisites: 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.)

Prerequisites: MATH*2160, MATH*3200

MATH*4070 Case Studies in Modelling 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 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: 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.)

Prerequisites: 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.

Prerequisites: 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.)

Prerequisites: MATH*2160, MATH*3200

MATH*4240 Advanced Topics in Modelling W(3-0) [0.50]

A study of selected advanced topics in mathematical modelling, 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.

Prerequisites: 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.

Prerequisites: 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; Reimannian geometry; tensor analysis; homotopy and homology groups. (Offered in odd-numbered years.)

Prerequisites: MATH*2160, 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.

Prerequisites: 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 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: MATH*3510 or MATH*3100, 0.50 credit in statistics


1999-2000 Undergraduate Calendar
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Last revised: January 1999.