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 ADVANCED MATHEMATICAL METHODS
Code MATH492
Coordinator Professor A Vogt
Mathematical Sciences
Andreas.Vogt@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2022-23 Level 7 FHEQ Second Semester 7.5

Aims

To give an introduction to the techniques of vector and tensor calculus and to the study and solution of the partial differential equations which arise in engineering.


Learning Outcomes

(LO1) After completing this module, the students should be familiar with the use of grad, div and curl and also the Kronecker delta and Levi-Civita tensor.

(LO2) They should understand the formulation and applications of Stokes' Theorem and the Divergence Theorem.

(LO3) They should be able to classify second-order partial differential equations and solve some standard examples in simple situations.

(S1) Problem solving skills

(S2) Numeracy


Syllabus

 

Introduction to tensor notation and its application to vector calculus Introduction to vector calculus: grad, div, curl and their use in differential geometry and in continuum mechanics, Line, surface and volume integrals, Stokes theorem, Divergence theorem and their applications to aerodynamics and fluid dynamics (e.g. circulation, vorticity) Classification of 2nd order PDEs: elliptic, hyperbolic, parabolic. Boundary conditions (Dirichlet, Neumann); uniqueness of solutions. Important examples: wave equation, heat equation, Laplace equation. Polar co-ordinates. PDEs for fluid motion discussed using grad, div and curl. Solutions eg using separation of variables, Fourier series or Fourier/Laplace transforms. Heat diffusion along a rod; vibrations of rectangular membranes.


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

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

 

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

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

 

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

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam There is a resit opportunity. Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Second semester    80       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Coursework Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Second semester    20