# Modules

### MODULES for Second-Year Course

Semester | Prerequisites | ||

2AC | Advanced Calculus | 1 | - |

2DE | Differential Equations | 2 | Modules 2AC and 2LA |

2FM* | Fourier Methods | 2 | - |

2IA* | Introductory Algebra | 2 | Module 2LA |

2LA* | Linear Algebra | 1 | - |

2RA* | Real Analysis | 2 | - |

* This module will not be offered in 2012.

**Syllabuses**

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.

2DS DISCRETE STRUCTURES:

Introduction to informal logic; use of truthtables, quantifiers. Methods of proof (contradiction, induction). Informal set theory, relations, equivalences, partitions, partial orders. Boolean algebras. Functions, cardinality. Recurrence relations. Introduction to graph theory.

2FM FOURIER METHODS:

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:

Further linear algebra: equivalence relations, the quotient of a vector space, the homomorphism theorem for vector spaces, direct sums, projections, nilpotent linear transformations, invariant subspaces, change of basis, the minimum polynomial of a linear transformation, unique factorization for polynomials, the primary decomposition theorem, the Cayley-Hamilton theorem, diagonalization. Group theory: subgroups, cosets, Lagrange's theorem, normal subgroups, quotients, homomorphisms of groups, abelian groups, cyclic groups, symmetric groups, dihedral groups, group actions, Caley's Theorem, Sylow's Theorems. Ring theory: rings, subrings, ideals, quotients, homomorphisms of rings, commutative rings with identity, integral domains, the ring of integers modulo n, polynomial rings, Euclidean domains, unique factorization domains. Field theory: subfields, constructions, finite fields, vector spaces over 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.