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 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 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 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, and vectors defined 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.