DoctorMichael Brideson

Andrew Roy Phair

Catherine Anne Browne

Aviva Samuelson

Rhys Alexander Paul

Jason Michael Cosgrove

David Edward Horsley

James Montagu Jackson

Peter Thomas While

This unit is a continuation of KMA152 and KMA154, with emphasis on the application of multivariable calculus and Fourier Series to problems in mathematics, the physical and biological sciences, economics, and engineering. The calculus section of this unit is focussed on dealing with functions of several variables, of which the typical case is z = f(x,y). Functions like this are important because they describe many of the situations we encounter when applying mathematics to models of the real world. The graph of the function z = f(x,y) is a surface, so it might be used to describe building structures; aeroplane wings; temperature, stress, and pressure distributions; income as a function of various expenses; and so on. We may need to say how rapidly such a surface curves, and that immediately requires us to do calculus on functions of two (or more) variables. We also need to consider vectors that are functions of several variables. Some obvious examples of these are the velocity vector in a moving fluid, the heat-flow vector in a solid, and the electric and magnetic fields produced by an antenna. This will lead us to consider more advanced concepts such as circulation, compressibility, divergence, and curl. Understanding these concepts is fundamental to the study of all areas of Engineering, Physics, and (continuum) Applied Mathematics. Topics will be introduced in the Cartesian (rectangular) coordinate system but we will also investigate functions, regions, and vectors defined in cylindrical and spherical coordinates. The Fourier-Series section of this unit is concerned with how to represent periodic functions as an infinite sum of the standard trigonometric functions, sine and cosine. This is an important concept for solving problems in acoustics, signal processing, heat-flow theory, fluid mechanics, vibrations, electromagnetic field theory, and so on.

Numerical methods are necessary in science and engineering, because most problems of practical interest are just too difficult to be solved in closed form. While many important problems, such as the motion of a mass on a spring etc., have exact solutions in terms of known functions such as sin or cos, these are nevertheless simple problems. Making these problems just a little more complicated leads very quickly to equations that have no easy solutions. Numerical techniques are therefore needed. Often the approach that is needed to solve a problem numerically is quite unlike the way that you would solve the same problem by hand (eigenvalues are a good example of this). In addition, the numerical solution of some problems can have its own behaviour, which may not be an accurate reflection of the behaviour of the true solution. For that reason, it is important to keep a check on how accurate a method is, and whether it converges (in some sense) to the true solution. This unit gives an introduction to using numerical methods to solve some of the key problems in science and engineering. We first consider how computers represent numbers and functions, then consider how to solve algebraic equations. A problem of major importance concerns the solution of linear (matrix) equations. Eigenvalues of a matrix are very important in applications, since they tell us the vibrational frequencies of a structure for example, and we will consider how to calculate the eigenvalues and eigenvectors of a matrix. Approximate methods for integrating and differentiating functions will be discussed. Finally, we will look at solving ordinary differential equations using the computer. Numerical principles will be demonstrated and explored with the scientific software Matlab.

This unit is primarily concerned with extending concepts of single variable calculus into the domain of several variables. It also looks at the construction of periodic functions with Fourier series. The calculus section of this unit is focussed on dealing with functions of several variables, of which the typical case is z = f(x,y). Functions like this are important because they describe many of the situations we encounter when applying mathematics to models of the real world. The graph of the function z = f(x,y) is a surface, so it might be used to describe building structures; aeroplane wings; temperature, stress, and pressure distributions; and so on. We need to be able to say how rapidly such a surface curves, and that immediately requires us to do calculus on functions of two (or more) variables. We will also need to consider vectors that are functions of several variables. Some obvious examples are the velocity vector in a moving fluid, the heat-flow vector in a solid, and the electric and magnetic fields produced by an antenna. This will lead us to consider more advanced concepts such as circulation, compressibility, divergence, and curl. Understanding this material is fundamental to the study of all areas of Engineering and (continuum) Applied Mathematics, and it underpins modern continuum mechanics and electromagnetic theory. Topics will be introduced in the Cartesian (rectangular) coordinate system but we will also investigate functions, regions, vectors, operations, and integration in cylindrical and spherical coordinates. The Fourier-Series section of this unit is concerned with how to represent periodic functions. Previously we have looked at power series as an infinite sum of terms involving increasing powers of a particular variable. A Fourier series is an infinite sum of terms involving sine and cosine functions. This is an important concept for solving problems in acoustics, signal processing, heat-flow theory, fluid mechanics, vibrations, electromagnetic field theory, and so on.

The unit has two major topic streams: computational techniques and probability and statistics. Probability and Statistics. An important aspect of engineering is the study, evaluation, and management of the reliability of systems, where reliability is defined as the ability of a system or a component to perform its required functions under stated conditions for a specified period. Reliability can be measured as a probability of failure, frequency of failures, or in terms of system availability. In this part of the unit we will give an introduction on statistics and the theory of probability, then concentrate on techniques of statistical analysis that are useful in engineering environments. Computational Techniques. The use of computers is ubiquitous in modern engineering with enormous faith put in the accuracy of their computations – is that faith founded? In this part of the unit we will explore fundamental numerical processes for obtaining approximate but useful solutions to mathematical models and operations. The consequence of adopting such approximations will be investigated through the evolution and propagation of errors. Statistical and numerical principles will be demonstrated and explored with scientific software including R and Matlab.

This unit is a continuation of KMA152, with emphasis on the application of single-variable calculus and linear algebra to problems in mathematics, the physical and biological sciences, economics, and engineering. Topics include complex numbers, linear equations and matrices, numerical integration and improper integrals, sequences and series, and ordinary differential equations. The main cohort will also explore parametric equations and polar coordinates while a maritime engineering cohort concentrates on statistics.The units KMA152 and KMA154 provide a substantial foundation and excellent introduction to the rigour required for higher level mathematics-based subjects.