# Mathematics

## ECM2706 - Vector Calculus and Applications (2015)

MODULE TITLE CREDIT VALUE Vector Calculus and Applications 15 ECM2706 Dr Joanne Mason (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 203
DESCRIPTION - summary of the module content

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

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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;
- irrotational and rotational motion;
- velocity potential for irrotational motion;
- vorticity, Bernoulli's equation;
- complex potential;
- uniform stream, sources, sinks and dipoles;
- vortex motion.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 44 106 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 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

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
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

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
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

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
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

RE-ASSESSMENT NOTES

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.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk