Vector Calculus and Applications - 2019 entry

This module introduces you to vector calculus and its applications, especially fluid dynamics, in science and engineering. It consists of two parts, which are closely linked. In the first part of the module, you will learn about the mathematical theory and techniques of vector calculus. You will become competent in using vector calculus in both differential and integral forms. The second part of the module gives an introduction to fluid dynamics as an application of vector calculus. It lays down some basic principles using a number of simplifying assumptions, in particular assumptions of inviscid, incompressible flow. Applications include the design of wind turbine blades and tidal barrages, the forces exerted by water of marine structures, and the motion of swimming organisms. These problems raise important questions, such as: How can we maximize power extracted. How do vortices form? What is pressure and how does it interact with the flow?

Pre-requisite modules: “Calculus and Geometry” (ECM1901); “Advanced Calculus” (ECM1905); and “Differential Equations” (ECM2903), or equivalent

This module aims to develop your understanding of fluid dynamics and motion of solids. It examines how one can use vector formalism and calculus together to describe and solve many problems in two and three dimensions. For example, the rules that govern the flow of fluids and the motion of solids can be described using vector calculus, with resulting laws of motion described by partial differential equations rather than ordinary differential equations.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Comprehend the meaning and use of vector calculus notation;

2 Perform manipulations with vector calculus in both differential and integral forms (line, surface and volume integrals);

3 Appreciate the application of vector calculus to problems in inviscid fluid mechanics;

Discipline Specific Skills and Knowledge:

4 Understand a number of mathematical modelling techniques for applications of fluid dynamics in engineering, biology and environmental science;

Personal and Key Transferable/Employment Skills and Knowledge:

5 Reveal how to formulate and solve complex problems;

6 Demonstrate appropriate use of learning resources;

7 Demonstrate self-management and time management skills.

- Summation convention;

- Definitions of scalar field, level surface, vector fields, field lines;

- Motivation from fluid flow;

- Vector differentiation and the differential operators: gradient, divergence, and curl;

- Examples in 3D for Cartesian, cylindrical and spherical coordinates;

- Line integrals and elementary surface and volume integrals;

- Stokes' theorem and the divergence theorem;

- Introduction to continuum mechanics and Eulerian fluid mechanics;

- Velocity, acceleration, streamlines and pathlines;

- The continuity equation and incompressibility;

- Gradient, divergence and curl in cylindrical and spherical coordinates;

- Vorticity and circulation;

- Pressure, constitutive equations, Euler's equations, steady and unsteady flows;

- Irrotational and rotational motion;

- Velocity potential for irrotational motion;

- Vorticity, Bernoulli's equation;

- Complex potential;

- Uniform stream, sources, sinks and dipoles;

- Vortex motion.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE: http://vle.exeter.ac.uk

Reading list for this module: