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## ECM2706 - Vector Calculus and Applications (2015)

MODULE TITLE | Vector Calculus and Applications | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM2706 | MODULE CONVENER | Dr Joanne Mason (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 weeks | 0 |

Number of Students Taking Module (anticipated) | 203 |
---|

This module introduces you to vector calculus and its applications, especially fluid dynamics. 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 as an application 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.

Applications include the design of aeroplanes, car body shapes and the flows of liquids and gases through pipes. These problems raise important questions, such as: How is flight possible? How can one minimise drag? How do vortices form? What is pressure and how does it interact with the flow? Physical applications include meteorology (fluid dynamics applied to weather forecasting and events such as tornadoes and hurricanes) and oceanography (fluid dynamics applied to ocean currents, tides and waves). This module is a prerequisite for a number of more specialist modules in the third year.

Prerequisite module: ECM1701 and ECM2702 or equivalent

This introductory vector calculus course aims to increase 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 with application to fluid dynamics.

**Personal and Key Transferable/ Employment Skills and Knowledge:**

5 reveal 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;

- 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.

Scheduled Learning & Teaching Activities | 44.00 | Guided Independent Study | 106.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures including examples classes |

Scheduled learning and teaching activities | 11 | Tutorials |

Guided independent study | 106 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Fortnightly exercise | 6-8 questions, 3-4 hours | 1, 2, 3, 4, 5 | Oral feedback in tutorial classes |

Coursework | 20 | Written Exams | 80 | Practical Exams |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours | 1, 2, 3, 4, 5 | Oral feedback on request |

Coursework – based on questions submitted for formative assessment | 20 | 5-7 questions, 3 hours | 1, 2, 3, 4, 5 | Written feedback by marker, oral feedback on request |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

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.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Acheson D.J. | Elementary Fluid Dynamics | Clarendon Press | 1990 | 978-0-198-59679-0 | [Library] | |

Extended | Batchelor G.K. | An Introduction to Fluid Dynamics | Cambridge University Press | 1999 | 000-0-521-04118-X | [Library] | |

Extended | Arfken G.B. & Weber H.J. | Mathematical Methods for Physicists | Electronic | Harcourt/ Academic Press | 2005 | 000-0-120-59825-6 | [Library] |

Extended | Tritton D.J. | Physical Fluid Dynamics | 2nd | Clarendon Press, Oxford | 1988 | 000-0-198-54493-6 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2702, ECM1701 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Friday 09 January 2015 | LAST REVISION DATE | Monday 12 January 2015 |

KEY WORDS SEARCH | Vector calculus; differential operators; line, surface and volume integrals; integral theorems; curvilinear coordinates; inviscid fluid dynamics. |
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