Bachelor of Arts (BA) - The University of Hong Kong

260MATH2301.Algebra I (6 credits)Group: Examples of groups, subgroups, coset, Lagrange theorem, quotient group, normal subgroup,group homomorphism, direct product of groups. Ring: Examples of rings, integral domain, ideal,Chinese Remainder theorem. Field of fractions, principal ideal domains, euclidean domains, uniquefactorization domains. Field: Definition and examples of fields. Polynomials: Polynomial ring in onevariable over a field, polynomials over integers, Gauss' lemma.Pre-requisite: 1. (Two out of MATH1101, MATH1102, MATH1201, MATH1202, one of whichshould be MATH1202); or2. (MATH1811/MATH1812 or MATH1803).MATH2401.Analysis I (6 credits)The metric topology of R n . Uniform convergence. Derivative of a function of several variables. Inversefunction theorem. Implicit function Theorem. Lagrange multiplier method.Pre-requisite: 1. (MATH1201 and MATH1202); or2. (MATH1811/MATH1812 or MATH1803).MATH2402.Analysis II (6 credits)Lebesgue integral of functions of one variable. Convergence Theorems. Integration of functions ofseveral variables. Fubini's theorem. Change of variables. Fields and forms. Poincaré lemma. Stokes'theorem.Pre-requisite: 1. (MATH1201 and MATH1202) and (MATH1101 or MATH1102); or2. (MATH1811/MATH1812 or MATH1803).MATH2403.Functions of a complex variable (6 credits)Complex number system. Analytic functions and elementary functions. The Cauchy-Riemannequations. Cauchy's theorem and its applications. Taylor's series. Laurent's series. Zeros, singularitiesand poles. The Residue Theorem and its applications.Pre-requisite: 1. (Two out of MATH1101, MATH1102, MATH1201, MATH1202, one of whichshould be MATH1201 or MATH1202); or2. (MATH1811/MATH1812 or MATH1803).MATH2405.Differential equations (6 credits)Review of elementary differential equations. Existence and uniqueness theorems. Second orderdifferential equations, Wronskian, variation of parameters. Power series method, Legendre polynomials,Bessel functions. The Laplace transform. Linear systems, autonomous systems. Qualitative propertiesof solutions.Pre-requisite: 1. (Two out of MATH1101, MATH1102, MATH1201, MATH1202, one of whichshould be MATH1201 or MATH1202); or2. (MATH1811/MATH1812 or MATH1803).MATH3302.Algebra II (6 credits)Introduction to module theory. Canonical forms of matrices: Rational canonical form, Jordancanonical form, invariant factor. Presentation of groups: Generators and relations, free group.Polynomial ring in several variables. Fundamental theorem on symmetric polynomials. Fieldsextension, elements of Galois theory (characteristic zero).Pre-requisite: MATH2301.