# Mathematics

## MTH2004 - Vector Calculus and Applications (2020)

MODULE TITLE CREDIT VALUE Vector Calculus and Applications 15 MTH2004 Dr Claire Foullon (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 250
DESCRIPTION - summary of the module content
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. Engaging applets, video examples and experiments are used throughout to illustrate the theory.

This module is a prerequisite for several specialist modules in the third year and fourth year, including MTH3007 Fluid Dynamics (which develops theory for flow with viscosity), MTH3001 Theory of Weather and Climate, MTHM031 Magnetic Fields and Fluid Flows and MTHM045 Space Weather and Plasmas.

Prerequisite modules: MTH2003, NSC1002 or equivalent
AIMS - intentions of the module

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.

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 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.

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;

- vorticity and circulation;

- 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
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 33 118
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching 33 Lectures including examples classes Scheduled learning and teaching activities 5 Tutorials Guided independent study 112 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
Exercise sheets 5 x 10 hours 1, 2, 3, 4, 5 Oral feedback in tutorial classes and drop-ins; written tutor feedback on submitted solutions.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 10 90
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 90 2 hours All Written/verbal on request, SRS.
Coursework classes 10 2 assignments, 30 hours total All Annotated script and written/verbal feedback

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

Type Author Title Edition Publisher Year ISBN Search
Set Finney, R.L., Maurice, D., Weir, M. and Giordano, F.R. Thomas' Calculus based on the original work by George B. Thomas, Jr. 10th or later Addison-Wesley 2003 000-0-321-11636-4 [Library]
Set Arfken, G.B. & Weber, H.J. Mathematical Methods for Physicists Electronic Harcourt/ Academic Press 2005 000-0-120-59825-6 [Library]
Set Acheson, D.J. Elementary Fluid Dynamics Clarendon Press 1990 978-0-198-59679-0 [Library]
Set Tritton D.J. Physical Fluid Dynamics 2nd Clarendon Press, Oxford 1988 000-0-198-54493-6 [Library]
Set Batchelor G.K. An Introduction to Fluid Dynamics Cambridge University Press 1999 000-0-521-04118-X [Library]
Set Matthews P.C Vector Calculus 1st Springer 1998 978-3540761808 [Library]
Set Spiegel M.R., Lipschutz S., Spellman D. Vector Analysis and an Introduction to Tensor Analysis 2nd McGraw-Hill 2009 9780071615457 [Library]
Set Paterson A.R. A first course in fluid dynamics Electronic Cambridge University Press 1983 9780521274241 [Library]
Set Nahvi, M; Edminister, J Schaum's Outline of Electromagnetics 5th McGraw-Hill Education 2019 9781260120974 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2003
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Tuesday 10 July 2018 Wednesday 29 July 2020
KEY WORDS SEARCH Vector calculus; differential operators; line, surface and volume integrals; integral theorems; curvilinear coordinates; inviscid fluid dynamics; electromagnetism