Pre-requisite: 64613/75613
This unit is broadly divided into two interrelated halves. The vector calculus half develops further the theory and applications of the calculus of vectors from that established in 64613. The principles of modelling material such as air or water are introduced in one-dimensional dynamics. Then the application of vector calculus to modelling the 3-D flow of a fluid material is developed along with the notions of scalar and vector functions, directional derivatives, divergence, curl and Laplace's equation. The integral theorems of Gauss and Stokes highlight many facets of fluid flow and other physical fields. Many practical problems involve curvilinear coordinates, requiring expressions for vector differential operators in general coordinate systems. An application is to flow in a thin, twisting tube. The differential equations (DE's) half of the course begins with a formal study of infinite series, leading to power series and Taylor's theorem n-dimensions. Application is made to the classification of extrema in constrained optimisation problems. The reduction of general DE's to systems of linear, first-order DE's is used to visualise dynamics using the phase diagram and to introduce fixed points, stability and limit cycles. Fourier analysis introduces the use of series expansion and integral transform methods which are then employed in the solution of partial differential equations that appear in models of many physical systems.