Functions of one real variable and their graphs. Differentiation, integration and differential equations with applications to mathematical models. Introduction to power series, numerical methods and partial differentiation.
Linear equations, lines and planes in two and three dimensions. Linear transformations, vectors, matrices and determinants in two and three dimensions, eigenvectors and eigenvalues. An introduction to linear programming and complex numbers.
160.103 Introductory University Mathematics15 credits
A course designed to increase the confidence of students in handling mathematical concepts and skills. Content includes algebraic skills, functions and graphs, and an introduction to matrices and calculus.
This course provides a solid mathematical foundation for further studies in mathematics, science and engineering. It consolidates basic concepts and introduces more advanced material on differentiation and numerical techniques, enabling the formation of mathematical models of real-world problems. The course blends topics from calculus with those from linear algebra and includes matrices, linear equations, vectors and geometry.
This course builds on the foundation provided by160.111. Together these courses provide a mathematical platform for more advanced studies in mathematics, science and engineering.. The topics are a blend of calculus and linear algebra, including complex numbers, linear transformations, eigenvectors, advanced techniques of integration, differential equations and applications.
Development of algebraic skills. An introduction to linear equations and matrices, including graphical linear programming. Graphs. An introduction to calculus. Use of spreadsheets and/or other mathematical software.
At the heart of this course are three mathematical questions: what is an equation, what is a solution and what is a function? Through exploring these three themes, students will be exposed to different types of equations, different types of solutions and mathematical functions. Students will also learn to differentiate, integrate and manipulate simple equations and develop problem solving skills.
A mathematical foundation for further studies in mathematics, statistics, natural and computing sciences, business and education. It combines a blend of concepts, techniques and applications. Topics from algebra and calculus include matrices, vectors and geometry, complex numbers, techniques and applications of differentiation and integration. The course follows from160.132; well-prepared students from high school can enter160.133 directly.
The techniques of 100-level calculus are applied and extended in the study of infinite series, vector-valued functions and functions of two or more variables. Topics include convergence of power series, partial derivatives, double and triple integrals with applications to surface area and volumes, line and surface integrals.
Exact solution methods for ordinary differential equations including the use of the Laplace transform. Systems of differential equations, matrix methods, phase plane techniques. Numerical methods for differential equations.
Real analysis: inequalities, the continuum property, induction, sequences, functions and limits, continuity, contraction mappings and fixed points, differentiation, mean value theorems and Taylor's theorem. Complex analysis: geometry in the complex plane, limits and continuity, holomorphic functions, line integrals, Cauchy's theorem and some elementary consequences, singularities and Laurent's theorem, the calculus of residues and some applications.
Group theory - basic properties, permutation groups, finite Abelian groups, cosets, normal subgroups, homomorphism theorems, representation. Ring theory - integral domains and fields, ideals, homomorphism theorems, factorisation, extension fields.
Ordinary differential equations: series solutions, special functions, Sturm-Liouville problems, Green's functions. Partial differential equations: method of characteristics, classification of second order equations, separation of variables, numerical methods, Fourier transforms.
The mathematical modelling process and methodologies examined through a variety of case studies. Application of analytical techniques, numerical methods and computer software packages to the solution of differential equations, difference equations and linear and nonlinear systems.
A selection of topics in advanced algebra which may include the following: isomorphism theorems, series of groups, Sylow theorems, classification of finitely generated abelian groups, free groups, group representations, matrix representations and characters of groups; extension fields, Galois correspondence, solvability of polynomial equations; semigroups, Green's equivalence, regular semigroups, inverse semigroups.
Advanced study of computational solution methods with topics selected from approximation theory, sparse linear systems, matrix eigenproblems, initial value problems and boundary value problems in ordinary differential equations and partial differential equations.
A selection of topics which may include asymptotic analysis, the calculus of variations, integral equations and partial differential equations. Some applications to problems in engineering and physics will be discussed.
160.734 Studies in Applied Differential Equations15 credits
Topics in the advanced study of ordinary and partial differential equations selected from dynamical systems, chaos, Lie symmetries, and applications to mathematical modelling, physics and engineering.
Studies of the mathematical formulation of the physical principles required for the development of modern theories in mathematical physics. A topic or topics will be selected from areas such as Lie groups and algebras, analytical mechanics, electrodynamics, quantum mechanics and kinetic theory, together with suitable applications.