### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title VECTOR CALCULUS WITH APPLICATIONS IN FLUID MECHANICS Code MATH225 Coordinator Dr DJ Colquitt Mathematical Sciences D.Colquitt@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2018-19 Level 5 FHEQ First Semester 15

### Aims

To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them.

To give an appreciation of the many applications of vector calculus to physical situations.

To provide an introduction to the subjects of fluid mechanics and electromagnetism.

### Learning Outcomes

After completing the module students should be able to:

-     Work confidently with different coordinate systems.

-     Evaluate line, surface and volume integrals.

-     Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes.

-     Recognise the many physical situations that involve the use of vector calculus.

-     Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow.

All learning outcomes are assessed by both examination and course work.

### Syllabus

Different coordinate systems.

Scalar and vector fields; electrostatic field, Lagrangian and Eulerian descriptions of a fluid.

Surface and volume integrals; divergence, Gauss'' theorem, equation of continuity, incompressible flows.

Curl, line integrals, Stokes'' theorem; irrotational fields, conservative fields,velocity potential.   Maxwell''s equations, wave equation, acceleration of a fluid particle.

Applications to fluid motion; Inviscid fluids, boundary conditions, pressure, Euler equation and solutions for irrotational motion and steady motion.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH102; MATH101

MATH326

### Programme(s) (including Year of Study) to which this module is available on a required basis:

Programme:F344 Year:2 Programme:FGH1 Year:2 Programme:FG31 Year:2

### Programme(s) (including Year of Study) to which this module is available on an optional basis:

Programme:G100 Year:2 Programme:G101 Year:2 Programme:G110 Year:2 Programme:G1N2 Year:2 Programme:G1R9 Year:2 Programme:G1X3 Year:2 Programme:GG13 Year:2 Programme:GL11 Year:2 Programme:GR11 Year:2 Programme:GN11 Year:2 Programme:GG14 Year:2 Programme:GV15 Year:2 Programme:G1F7 Year:2 Programme:BCG0 Year:2 Programme:Y001 Year:2 Programme:L000 Year:2

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Unseen Written Exam  2.5 hours  First semester  85  Yes  Standard UoL penalty applies  Assessment 2 Notes (applying to all assessments) 10% homework and 5% class test This work is not marked anonymously. Candidates should attempt all questions in Section A and three questions in Section B. Section A carries 55% of the available marks. A formula sheet is attached at the end of the paper.
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Coursework    First semester  15  None: exemption approved November 2007  Standard UoL penalty applies  Assessment 1