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Undergraduate Calendar Content 2008-2009 MATH MATHEMATICS 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 transcript. More information can be obtained from http://www.math.unb.ca or from the Department. 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. Page 1 of 7 Undergraduate Calendar Content 2008-2009 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. Page 2 of 7 Undergraduate Calendar Content 2008-2009 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 3113. 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. Page 3 of 7 Undergraduate Calendar Content 2008-2009 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. Page 4 of 7 Undergraduate Calendar Content 2008-2009 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 mathematics. 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. Page 5 of 7 Undergraduate Calendar Content 2008-2009 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. Page 6 of 7 Undergraduate Calendar Content 2008-2009 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 Department Page 7 of 7

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