Mathematics 2000

Course outline:
The aim of this course is to introduce the student to a selection of fundamental topics in mathematics. The course consists of four modules chosen from the list below, but with the module 2LA being compulsory. All students must take at least one of the modules 2IA or 2RA. Students who intend to proceed to MAM3000W should do both these modules. The modules listed below are included in this course (Note: all the modules may not be offered in any one year).

The syllabus covers the following topics:
2AC ADVANCED CALCULUS Differentiable functions, independence of order of repeated derivatives, chain rule, Taylor's theorem, maxima and minima, Lagrange multipliers. Curves and surfaces in three dimensions, change of coordinates, spherical and cylindrical coordinates. Line integrals, surface integrals. Stokes' theorem. Green's theorem, divergence theorem.

2DE DIFFERENTIAL EQUATIONS: This module is aimed at Actuarial and Business Science students. A selection from the following topics will be covered: First order difference equations. Second order difference equations with constant coefficients. Systems of first order difference equations. Linear differential equations and systems with constant coefficients. Laplace transforms and applications. Nonlinear equations and phase plane analysis. Parabolic partial differential equations, separation of variables, two point boundary value problems. Option pricing by the Black-Scholes equation. Stochastic Differential Equations. All topics will have applications to economics and finance.

2FM FOURIER METHODS (this module will not be offered in 2014): Signals and systems. Fourier series. Analysis of periodic Fourier series. Discrete frequency spectra. Fourier transforms, convolution, continuous spectra. Applications. Discrete and Fast Fourier Transforms.

2IA INTRODUCTORY ALGEBRA: Group theory: basic properties, subgroups, cosets, equivalence relations, Lagrange's theorem, order of an element, cyclic groups, generation of groups, permutation groups, parity, conjugation, cycle structure, normal subgroups, quotients, homomorphisms, group actions. Number theory: basic properties of the integers, unique factorization, congruences. Ring theory: subrings, ideals, integral domains, Euclidean domains, polynomial rings, application to linear algebra. Field theory: field of fractions, finite fields.

2LA LINEAR ALGEBRA Matrices, Gauss reduction, invertibility. Vector spaces, linear independence, spans, bases, row space, column space, null space. Linear maps. Eigenvectors and eigenvalues with applications. Inner product spaces, orthogonality.

2RA REAL ANALYSIS Sequences, subsequences, Cauchy sequences, completeness of the real numbers. Series: convergence, absolute convergence and tests for convergence. Continuity and differentiability of functions. Taylor series and indeterminate forms. Sequences and series of functions, uniform convergence, power series.