Mathematics 2A module (MA21001)

This module is part of a series of four modules, Mathematics 1A, 1B, 2A, and 2B, which are the core Mathematics modules in years 1 and 2. These modules provide a solid foundation in calculus, algebra, and geometry required for higher-level mathematics modules.   

The module consists of a calculus and an algebra component. In the calculus component, you will learn about the mathematical notions of continuity, differentiability and integrability of functions, which are the foundations for analysing many problems in physics, biology, economics and finance. In the algebra component, you will encounter the notion of a vector space and learn to calculate with matrices. These concepts have a wide range of applications, ranging from the theories of quantum mechanics and electrodynamics to computer science and optimisation in finance.  

Topics include 

• Differential equations - revision of linear differential equations of second order with constant coefficients using undetermined coefficients. General solutions and solutions satisfying initial conditions. Resonance. Equations of higher order.  
• Hyperbolic functions - hyperbolic functions: Solution of simple equations, inverse functions. Revision of standard methods of definite integration, including hyperbolic substitutions.  
• The fundamentals of calculus - limits, continuity, and differentiability. Rolle's theorem, mean value theorem. The definition and properties of the Riemann integral. The fundamental theorem of calculus. L′Hôpital′s rule and indeterminate forms. Infinite and improper integrals. Taylor and Maclaurin series.   
• Vectors and vector spaces - definition of a vector space. Vectors, lines and planes. Span, linear independence. Basis and dimension. Subspaces.  
• Inner product - scalar product, length. Projection. Normal form of hyperplanes. Orthogonality.  
• Linear equations and matrices - systems of linear equations. Gaussian elimination. Matrices and matrix operations, transposes and inverses. Matrix equations. LU factorisation. Determinants.