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MATH Undergraduate Calendar Content 2008 2009 MATH MATHEMATICS


									Undergraduate Calendar Content

See also "Statistics".

Credit for MATH 1003

    1. Calculus Challenge Exam

        This examination which is held in early June is open to students registered in a
        calculus course at a high school that has made arrangements with the Department of
        Mathematics & Statistics. A fee will be charged.

        Students who qualify for credit will receive a certificate entitling them to credit for and
        therefore exemption from MATH 1003 when they register at UNB. Upon the student's
        acceptance of the credit (3ch), the letter grade of the exam will be recorded on their

        More information can be obtained from or from the

    2. Advanced Placement Test

        The Science Faculty offers Advanced Placement Tests for some first year science
        courses, including MATH 1003, during registration week (early September) each year.

        More information can be obtained by consulting the Science section of the calendar or
        by contacting the Science Faculty or the Department of Mathematics & Statistics.

Note: All prerequisite courses must be passed with a grade of C or better. See beginning of
Section H for abbreviations, course numbers, and coding.

MATH 1003           Introduction to Calculus I                        3 ch (4C)
Functions and graphs, limits, derivatives of polynomial, log, exponential and trigonometric
functions. Curve sketching and extrema of functions. NOTE: Credit will not be given for both
MATH 1003 and 1823. Prerequisite: A minimum grade of 60% in New Brunswick high school
courses: Trigonometry and 3-space, Advanced Math with an Introduction to Calculus, or
equivalent courses; and a passing score on the Department of Mathematics and Statistics
placement test.

MATH 1013             Introduction to Calculus II                      3 ch (4C)
Definition of the integral, fundamental theorem of Calculus, Techniques of integration,
improper integrals. Ordinary differential equations. Taylor polynomials and series.
Prerequisite: A grade of C or higher in MATH 1003.

MATH 1053            Enriched Introduction to Calculus                   3 ch (4C)
The syllabus is similar to that for MATH 1003, with more emphasis placed both on the theory
of Calculus and interesting applications. The course will be of special interest to students with
strong Mathematical backgrounds. Any interested student (with or without High School
Calculus) is encouraged to consult with the Mathematics Department. Prerequisite: A grade of
85% or higher in a Grade 12 Math course that contains some Calculus, or consent of the
Mathematics Department.

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Undergraduate Calendar Content
MATH 1063            Enriched Introduction to Calculus II              4 ch (4C)
The syllabus for this course is similar to that of MATH 1013. As with MATH 1053, more
emphasis is placed on theory, mathematical rigor and interesting applications. Prerequisite: A
grade of B or higher in MATH 1053.

MATH 1503             Introduction to Linear Algebra                 3 ch (3C)
Lines and Planes, The Geometry and Algebra of vectors, Systems of linear equations, Matrix
Algebra, Linear Independance, Linear Transformations, Determinants, Complex numbers,
Eigenvalues, Eigenvectors, Diagonalization, Rotation matrices, Quadratic forms, Least squares.
Prerequisite: A minimum grade of 60% in New Brunswick high school courses: Trigonometry
and 3-space, Advanced Math with an Introduction to Calculus, or equivalent courses. Note:
Credit will not be given for both Math 1503 and Math 2213.

MATH 1823           Calculus for Management Sciences                   3 ch (3C 1T)
Polynomial, logarithmic and exponential functions. Limits and derivatives. Extreme values and
related rates. Simple integration. Differential equations. Throughout stresses applications to
business and economics. NOTE: Credit will not be given for both MATH 1003 and 1823.
Prerequisite: A minimum grade of 60% in New Brunswick high school courses: Trigonometry
and 3-space, Advanced Math with an Introduction to Calculus, or equivalent courses.

MATH 1833            Finite Mathematics for Management
                     Sciences                                        3 ch (3C)
Matrices and systems of linear equations. Linear programming concepts; graphical solution of
two variable problems. Permutations and combinations. Elementary probability. Mathematics
of finance. NOTE: Credit for MATH1833 will not be given if the student has previously taken
either MATH 1503 or MATH 2213. Prerequisite: New Brunswick Mathematics 112 GA
(Geometry and Applications) and New Brunswick Mathematics 112 FR (Functions and
Relations), or equivalent.

MATH 2003            Intermediate Mathematics I                         3 ch (3C 1T)
Analytic geometry and vectors. Parametric curves. Polar, cylindrical and spherical coordinates.
Functions of several variables, partial derivatives, applications to max-min. Double and triple
integrals. Prerequisite: A grade of C or higher in MATH 1013 or MATH 1063.

MATH 2013            Intermediate Mathematics II                        3 ch (3C 1T)
Review of first order differential equations. Second order linear O.D.E.'s. Infinite series,
including power series solutions to O.D.E.'s. Line and surface integrals. Theorems of Green
and Stokes. Divergence Theorem. Prerequisite: A grade of C or higher in MATH 2003.

MATH 2203            Discrete Mathematics                             3 ch (3C)
Logic, methods of proof, mathematical induction, elementary set theory, functions and
relations. NOTE: This course is designed for students desiring a good grounding in the
foundations of mathematics. Theorems and proofs are an important part of the course. Credit
will not be given for both MATH 2203 and CS 1303. Students majoring in Mathematics must
take MATH 2203. Prerequisite: MATH 1063 or MATH 1013 or permission of instructor. NOTE: It
is strongly recommended that students should have at least a grade of B in MATH 1013 to
take this course.

MATH 2213             Linear Algebra I                                 3 ch (3C)
Linear equations, matrix algebra, determinants, vector spaces, basis, row and column spaces,
linear transformations and matrix representations, scalar products, orthogonal projection,
least squares, eigenvectors and diagonalization, quadratic forms, singular value
decomposition. The course will include use of mathematical software. Prerequisite: MATH
1013, or MATH 1053, or both MATH 1823 and 1833. This course may also be taken with the
consent of the instructor. Interested first year students are encouraged to enquire. Note:
Credit will not be given for both Math 1503 and Math 2213.

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Undergraduate Calendar Content
MATH 2513           Multivariable Calculus for Engineers                  4 ch (4C)
Functions of several variables, partial derivatives, multiple integrals, vector functions, Green's
and Stokes' Theorems. See the note following MATH 2003. Prerequisite: A grade of C or higher
in both MATH 1013 and MATH 1503.

MATH 2633            Fundamental Principles of Elementary
                     School Mathematics                                 3 ch (3C 1L)
This course is intended for students who anticipate a career as an elementary teacher. The
course focuses on the mathematical content with topics taken from the K-6 Atlantic Canada
Mathematics Curriculum and extensions beyond the classroom to show the how and why
behind school mathematics. The major topics are problem solving, number concepts, number
and relationship operations, patterns and relations, shape and space, as well as data
management and probability. Intended for students registered in concurrent education or arts
programs. Not available for credit to students registered in the following programs:
Mathematics (honours, major, or minor), Statistics (honours, major, or minor), Computer
Science, Engineering, Administration. Prerequisite: Successful completion of at least one year
of a university program, and consent of the undergraduate advisor for mathematics.

MATH 3003            Applied Analysis                                  3 ch (3C)
Vector spaces of functions, convergence in normed linear spaces, orthogonal polynomials,
Fourier series, Fourier transform, Fast Fourier transform, introduction to wavelets, and
selected applications. Prerequisites: MATH 2013 or MATH 3503, and MATH 2213 or MATH 1503
(MATH 3213 recommended). NOTE: Credit will not be given for both MATH 3003 and MATH

MATH 3033          Group Theory                                      3 ch (3C)
Groups are the mathematical objects used to describe symmetries. This course covers the
fundamentals of group theory, together with applications selected from chemistry, geometry
and advanced algebra. Prerequisites: Either MATH 2203 or CS 2303, and MATH 2213 or MATH
1503 (MATH 3213 recommended).

MATH 3043             Nonlinear Differential Equations,
                      Stability and Chaos                               3 ch (3C)
Many of the processes studied in science, engineering and economics are nonlinear. This
course covers geometrical, analytical and numerical methods for systems of nonlinear ordinary
differential equations as an introduction to nonlinear phenomena: stability, attractors,
bifurcation and chaos. Also covered are the basic local existence and uniqueness theorem and
its applications, as well as linear systems and nonlinear difference systems to the extent
necessary to understand approximations to nonlinear differential equations. An introduction to
the use of mathematical software to illustrate regular and chaotic behaviour is included.
Prerequisite: MATH 2013 or both MATH 2513 and 3503.

MATH 3063            Geometry                                      3 ch (3C)
Axiomatic systems, non-Euclidian geometry, transformations in geometries, topological
properties of figures. Recommended for Education students or prospective Mathematics
teachers. Prerequisite: 9 ch in Math and/or Stat.

MATH 3073            Partial Differential Equations                      3 ch (3C)
Methods of solution for first order equations. Classification of second order equations.
Characteristics. Analytic and numerical methods of solution for hyperbolic, elliptic and
parabolic equations. Prerequisite: MATH 2013 or both MATH 2513 and 3503.

MATH 3093           Elementary Number Theory                        3 ch (3C)
Primes, unique factorization, congruences, Diophantine equations, basic number theoretic
functions. Recommended for Education students or prospective Mathematics teachers.

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Undergraduate Calendar Content
MATH 3103           Analysis I                                     3 ch (3C)
The real number system. Elementary set theory. Metric spaces. Sequences and series.
Continuity. Prerequisites: MATH 2013, 2203, and MATH 2213 or 1503.

MATH 3113             Analysis II                                       3 ch (3C)
Differential calculus, integration, sequences and series of functions, completeness of basis,
convergence of Fourier Series, Fourier Transforms, wavelets and wavelet transforms.
Prerequisite: MATH 3103. NOTE: Credit will not be given for both MATH 3003 and MATH 3113.

MATH 3213            Linear Algebra II                               3 ch (3C)
Possible topics: Vector spaces and subspaces, independent and spanning sets, dimension,
linear operators, determinants, inner product spaces, canonical forms. Prerequisite: MATH
2213 or MATH 1503 or consent of the instructor.

MATH 3243           Complex Analysis                                 3 ch (3C)
Complex analytic functions, contour integrals and Cauchy's theorems; Taylor's, Laurent's and
Liouville's theorems; residue calculus. Prerequisites: MATH 2003, MATH 2013 or equivalent.

MATH 3333            Combinatorial Theory                               3 ch (3C)
Topics selected from: Principle of inclusion and exclusion, Mobius inversion, generating
functions; systems of distinct representatives, Ramsey's Theorem; duality in external
problems, duality in programing; dynamic programing; block designs; introduction to matroid
theory; signal-flow graphs. (The course is also of interest to students in Computer Science and
Engineering.) Prerequisite: MATH 1003, 1823 or 1833.

MATH 3343           Networks and Graphs                               3 ch (3C)
Graphs, Euler paths, tournaments, factors, spanning trees, applications; electric networks and
Kirchhoff's laws, matroids; kernels, Grundy function and application to game theory; Menger's
theorem, flows in networks, flow algorithms. Prerequisite: MATH 1003, 1823 or 1833.

MATH 3353            Computational Algebra                              3 ch (3C)
Topics in abstract algebra are approached from the perspective of what can be computed
using such software packages as Maple, Macaulay and GAP. The topics covered will be
selected from: Grobner bases, resultants, solving polynomial equations, invariant theory of
finite groups, and the exact solution of differential equations. The course work will include a
mixture of problem sets emphasizing theory and pratical lab assignments. Prerequisites: one
of MATH 1013 or MATH 1063, and one of MATH 1503 or MATH 2213.

MATH 3363            Finite Mathematics (A)                           3 ch (3C)
Applications of algebraic and combinatorial methods to a selection of problems from coding
theory, computability, information theory, formal languages, cybernetics and the social and
physical sciences. Prerequisite: 12 ch in Math and/or Stat.

MATH 3413            Introduction to Numerical Methods                  4 ch (3C)
Error analysis, convergence and stability. Approximation of functions by polynomials.
Numerical quadrature and differentiation. The solution of linear and nonlinear equations and
the solution of ordinary differential equations. This course will emphasize the development of
computer algorithms and stress applications in the applied sciences. Note: This course is also
listed as CS 3113. Credit will not be given for both MATH 3413 and CS 3113. Prerequisites: CS
1003 or CS 1073, and MATH 2213 or MATH 1503.

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Undergraduate Calendar Content
MATH 3473             Mathematical Models (A)                           3 ch (3C)
Overview of the field of mathematical biology. Development, simulation and analysis of simple
mathematical models describing biological systems. Equal emphasis is placed on developing
simple models and case studies of successful models. The principle mathematical tools are
differential and difference equations, finite mathematics, probability and statistics. Note: This
course is also listed as BIOL 4563. Projects and assignments for MATH 3473 will place more
emphasis on model development and analysis. Students cannot receive credit for both BIOL
4563 and MATH 3473. Prerequisite: a statistics course, MATH2013 or MATH2513 or permission
of the instructor.

MATH 3503            Differential Equations for Engineers             3 ch (3C 1T)
Nonhomogeneous differential equations, undetermined coefficients, variation of parameters,
systems of 1st and 2nd order ordinary differential equations, Laplace transforms, Fourier
series. Prerequisite: MATH 1503 or 2213 (C grade minimum). Co-requisite MATH 2513 or
MATH 2003.

MATH 3543           Differential Geometry for Geomatics
                    Engineers                                        4 ch (4L 1T)
Basic analytic geometry, spherical trigonometry, geometry of curves in space, measurements
on surfaces, Gaussian surface geometry. Prerequisites: MATH 2513.

MATH 3623           History of Mathematics (A)                      3 ch (3C) [W]
A non-technical survey of the development of mathematics from primitive peoples through
Indian, Oriental, Babylonian, Egyptian and Greek cultures. More emphasis will be placed on
Western European and post-Renaissance mathematics, and recent (post-1940) history. An
attempt is made to discuss each new mathematical contribution in light of both past
mathematics and social scientific forces of the day. Some background in Mathematics
necessary. Prerequisite: 12 ch in Math and/or Stat.

MATH 3633           Fundamental Principles of School
                    Mathematics I                                    3 ch (3C)
A course for undergraduate students who anticipate a career as teachers. Topics build around
the K-12 syllabus, with extensions beyond the classroom, to show the 'how' and 'why' behind
school mathematics. Mathematical language; real numbers and other mathematical
structures; Euclidean geometry; functions; mathematical connections; problem solving.
Intended for students registered in concurrent B.Ed. programs, but may be taken by others
with the approval of the student's departmental Chair or Dean. Prerequisite: 6 ch of university

MATH 3803          Introduction to the Mathematics of Finance 3 ch (3C)
Measurement of interest, compound interest, annuities, amortization schedules and sinking
funds. Bonds. Prerequisite: MATH1013 or a grade of B or better in MATH 1823.

MATH 3813            Mathematics of Finance II (0)                   3 ch (3C)
A more advanced study of the topics in MATH3803 including varying and continuous annuities
and yield rates. Prerequisite: MATH3803 with a grade of B or better.

MATH 3843            Introduction to Life Contingencies                   3 ch (3C)
Survival distributions, general life insurances and life annuities, reserves. Joint annuities and
last survivor annuities. Prerequisite: One term of statistics and MATH3803.

MATH 4023           Functional Analysis                            3 ch (3C)
Normed spaces, the Hahn-Banach theorem, uniform boundedness theorem. Wavelets. The
contraction mapping theorem. Existence and uniqueness for nonlinear differental equations.
Prerequisite: Any two of MATH 3003, 3103, 3113, or permission of the instructor.

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Undergraduate Calendar Content
MATH 4043           Advanced Algebra (A)                                 3 ch (3C)
Prime fields and characteristic, extension fields, algebraic extensions, theory of finite fields,
Galois theory, and topics which may include some of: rings, topological algebra, multilinear
and exterior algebra, quadratic forms. Prerequisites: MATH 3033.

MATH 4063           Advanced Geometry (Exotic Spaces) (O)              3 ch (3C)
A deeper investigation of Euclidean and Non-Euclidean spaces of any dimension. Topics
selected from: axiom systems, linear and affine transformations, conformal and linear models
for Euclidean and hyperbolic spaces and their isometry groups, basic theory of convexity,
combinatorial properties of polytopes. Prerequisites: At least one of MATH 2213 or MATH 2003
or MATH 2513 or MATH 3063.

MATH 4100           Honours Project                                    6 ch [W]
Mathematics Honours students must complete a project under the supervision of a faculty
member. The project is to include a written report and an oral presentation. Prior to being
admitted into MATH 4100, the student must have been admitted to the Honours Program and
have submitted an acceptable project proposal to the department. Normally students would
begin preparation and research for the project during their third year of study, submit the
proposal by October of their fourth (final) year of study, and complete the written and oral
presentation by the end of the winter term, to graduate in May of that year.

MATH 4123           Advanced Linear Algebra (O)                      3 ch (3C)
The theory of vector spaces and linear transformations, dual spaces, multilinear maps
(including tensors and determinants); further topics chosen from canonical forms, metric
vector spaces, algebras, etc. Prerequisites: MATH 3213.

MATH 4153           Topology (A)                                     3 ch (3C)
A continuation of the topological concepts introduced in MATH 3103. Basic results in point-set
topology. Prerequisites: MATH 3103.

MATH 4413             Fluid Mechanics (A)                              3 ch (3C)
Derivation of the Equations of Motion: Euler's equations, rotation and vorticity, Navier-Stokes
equations. Potential Flow: complex potentials, harmonic functions, conformal mapping,
potential flow in three dimensions. Slightly Viscous Flow: boundary layers and Prandtl
boundary layer equations. Gas Flow in one dimension: characteristics and shocks.
Prerequisite: MATH 2003-2013 or equivalent.

MATH 4423             Mathematical Theory of Control (A)               3 ch (3C)
Topics selected according to the interests of students and faculty which may include the
following: optimal control of linear systems, Pontryagin's maximum principle, controlability,
observability, distributed parameter systems, differential games, stochastic systems.
Prerequisite: MATH 2003-2013 or equivalent.

MATH 4433           Calculus of Variations (A)                          3 ch (3C)
Introduction to functionals and function spaces. Variation of a functional. Euler's equations,
necessary condition for an extremum, case of several variables, invariance of Euler's equation,
fixed end point problem for unknown functions, variational problems in parametric form,
functionals depending on high order derivatives. Prerequisite: MATH 2003-2013 or equivalent.

MATH 4443            Introduction to Quantum Field Theory             3 ch (3C)
Relativistic quantum mechanics. The negative energy problem. Classical field theory,
symmetries and Noether's theorem. Free field theory and Fock space quantization. The
interacting field: LSZ reduction formula, Wick's theorem, Green's functions, and Feynman
diagrams. Introduction to Quantum electrodynamics and renormalization. This course is
crosslisted as PHYS 5153. Prerequisites: MATH 3003, PHYS 3051, and one of MATH 3043,
3503, PHYS 3011, 3031, or permission of instructor.

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Undergraduate Calendar Content
MATH 4453            Special Functions (A)                             3 ch (3C)
Covers in depth those functions which commonly occur in Physics and Engineering, namely,
the Gamma, Beta, Bessel, Legendre, hypergeometric, Hermite and Laguerre functions.
Additional or alternative special functions may be included. Applications to Physics and
Engineering will be discussed. Prerequisite: MATH 3043 or 3503 or equivalent.

MATH 4473          Introduction to Differential Geometry (A)     3 ch (3C)
Geometry of embedded curves and surfaces, n-dimensional manifolds, tensors, Riemannian
geometry. Prerequisites: MATH 2003-2013 or equivalent.

MATH 4483             Introduction to General Relativity (A)               3 ch (3C)
Special relativity, foundations of general relativity, solutions of Einstein's equations, classical
tests, cosmology, additional topics. Prerequisites: MATH 4473 or consent of instructor.

MATH 4503             Numerical Methods for Differential Equations 3 ch (3C)
The numerical solution of ordinary differential equations, and partial differential equations of
elliptic, hyperbolic and parabolic type. The course is a basic introduction to finite difference
methods, including the associated theory of stability, accuracy and convergence. Students will
gain practical experience using state-of-the-art numerical solvers and visualization tools, while
solving problems from the physical and biological sciences. Prerequisites: One of: MATH 3043,
3073, 3503, CS 3113, CHE 3418, or ME 3522.

MATH 4633            Calculus Revisited                                3 ch (3C)
A course for high school mathematics teachers. The course is built around a set of
optimization problems, whose solution requires review of topics in first and second year
calculus and linear algebra. Connections are made with topics in the Common Atlantic High
School Mathematics Curriculum. Prerequisite: Permission of Instructor. Students should be
near completion of requirements for a major or minor in mathematics.

MATH 4643             Formal Languages                                   3 ch (3C)
Brief history of structural linguistics. Introduction to mathematical methods of linguistics.
Finite state automata, regular languages. Computability. Chomsky hierarchy. Phrase-structure
grammars. Artificial intelligence problem. Critiques of structural linguistics. Prerequisite:
Consent of the instructor. MATH 2203 or CS2303 recommended.

MATH 4853            Mathematics of Financial Derivatives (A)            3 ch (3C)
Basics of options, futures, and other derivative securities. Introduction to Arbitrage. Brief
introduction to partial differential equations. Stochastic calculus and Ito's Lemma. Option
pricing using the Black-Scholes model. Put-call parity and Hedging. Pricing of European and
American call and put options. Numerical methods for the Black-Scholes model: binary trees,
moving boundary problems, and linear complementarity. The barrier, and other exotic options.
Prerequisites: (MATH 3503 and STAT 2593) or (MATH 2013, 2213 and STAT 3083), and
CS1073 or experience with a computer programming language.

MATH 4903           Independent Study in Mathematics                   3 ch
Topics to be chosen jointly by student, advisor, and Department Chair. May be taken for credit
more than once. Title of topic chosen will appear on transcript. Prerequisite: Permission of

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