Vector Calculus and Applications - 2019 entry

This module introduces you to vector calculus and its applications, especially fluid dynamics but also electromagnetism. 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 develop your competence in using vector calculus in both differential and integral forms. The second part of the module gives an introduction to fluid dynamics and electromagnetism as two applications of vector calculus. It lays down some basic principles using a number of simplifying assumptions. The module will emphasise inviscid, incompressible flow; later modules will cover the subject of viscous flow.

This module is a prerequisite for a number of more specialist modules in the third year. 



Prerequisite modules: MTH1002 or NSC1002 and MTH2003 or equivalent.

This introductory vector calculus course aims to increase your understanding of fluid dynamics and electromagnetism. 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 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 define and express vector calculus notation;

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

3 implement and illustrate the application of vector calculus to problems in inviscid fluid mechanics and electromagnetism.


Discipline Specific Skills and Knowledge:

4 validate a number of mathematical modelling techniques with application to fluid dynamics and electromagnetism.


Personal and Key Transferable/ Employment Skills and  Knowledge:

5 devise how to formulate and solve complex problems.

- 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;

- vorticity and circulation;

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

- irrotational and rotational motion;

- velocity potential for irrotational motion;

- Bernoulli's equation;

- Streamlines, Vortex lines and the Stream Function;

- Charge conservation

- Maxwell’s equations

- Electromagnetic potentials 

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.

ELE – http://vle.exeter.ac.uk

Reading list for this module: