Maths modules - First year
Real Analysis and the Calculus
The calculus is one of mankind’s most significant scientific achievements, transforming previously intractable physical problems into often routine calculations. Although its roots trace back into antiquity, it was developed in the late 17th century by Newton, when developing his laws of motion and gravitation, and Leibniz, who developed the notation we still use today. Analysis is the branch of mathematics that underpins the theory behind the calculus, placing it on a firm logical foundation through the introduction of the notion of a limit. This module introduces differentiation and integration from this rigorous point of view. The notion of a function of a real variable and its derivative are formalized. The familiar techniques and applications of differentiation and integration are reviewed and extended. Simple first and second order ordinary differential equations are studied. The theory of infinite sequences and series, including Taylor series, is introduced.
Vectors, Geometry and Linear Algebra
This module introduces a number of powerful ideas found in all area of mathematics and its applications that are broadly geometric in flavour. Complex numbers, which turn out to underpin a profound unification of many ideas in mathematics, are introduced. Vectors, which have both magnitude and direction, provide a natural way to describe lines and planes and are the appropriate language with which to model physical systems in mechanics. Matrices provide both a convenient way to deal with large systems of linear equations and to transform vectors and coordinate systems. This in turn leads to the theory of linear algebra and vector spaces. The abstraction of the notion of a vector space is another powerful unifying theory that is found across mathematics, with applications in abstract group theory, video games and signal processing. Coordinate systems for the Euclidean plane are discussed and the standard theory of conic sections is developed. The module also introduces the fundamental proof technique of Mathematical Induction.
Classical or Newtonian mechanics is the foundation of applied mathematics and is an astonishingly powerful tool for explaining physical systems, from projectiles to planetary motion to the design of racing cars. It acts as a natural starting point for any serious discussion of mathematical modelling in broader areas. This module uses ideas such as forces, moments, Newton's Laws of Motion and energy to model practical situations. These models can then be analysed using a wide range of techniques from pure mathematics such as trigonometry, algebra, calculus and, in particular, vector methods. Real world problems are used to illustrate the theory and some surprising and counter-intuitive examples are discussed.