Mathematics

MTH2004 - Vector Calculus and Applications (2022)

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MODULE TITLEVector Calculus and Applications CREDIT VALUE15
MODULE CODEMTH2004 MODULE CONVENERDr 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), MTH3008 Partial Differential Equations, MTH3001 Theory of Weather and Climate, MTHM031 Magnetic Fields and Fluid Flows and MTHM045 Space Weather and Plasmas.
 
Prerequisite modules: MTH1002, 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;
 
- 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
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33.00 Guided Independent Study 118.00 Placement / Study Abroad
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 Written/ oral feedback in tutorial classes and drop-ins; written/oral feedback on submitted solutions.
Online Quizzes 3x (8-30) min 1,2 Written/oral feedback online and in class. 

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 40 Written Exams 60 Practical Exams
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 60 1.5 hours (January) 1,2,3,4 Written/verbal on request, SRS.
Coursework 1 10 10 hours  All Annotated script and written/verbal feedback online and in class
Coursework 2 30 20 hours 1,2,3,4 Annotated script and written/verbal feedback online and in class 
         

 

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

Reassessment will be by a single written exam only, worth 100% of the module. For referred candidates, the mark will be capped at 40%. For deferred candidates, the mark will be uncapped.

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

Reading list for this module:

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 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES None
CO-REQUISITE MODULES None
NQF LEVEL (FHEQ) 5 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Tuesday 10 July 2018 LAST REVISION DATE Friday 05 August 2022
KEY WORDS SEARCH Vector calculus; differential operators; line, surface and volume integrals; integral theorems; curvilinear coordinates; inviscid fluid dynamics; electromagnetism