UNIVERSITY OF KENT
Confirmation that this version of the module specification has been approved by the School
Learning and Teaching Committee:
……13th May 2015…………………….(date)
MODULE SPECIFICATION
1.
Title of the module: Applied Differential Geometry (MA609 and MA969)
2.
School or partner institution which will be responsible for management of the
module: School of Mathematics, Statistics and Actuarial Science
3.
Start date of the module: MA609 September 2015; MA969 Autumn 2010 (revised version
start date September 2015)
4.
The number of students expected to take the module: 30
5.
Modules to be withdrawn on the introduction of this proposed module and
consultation with other relevant Schools and Faculties regarding the withdrawal
MA969 is an existing module. MA609 is a new module that focusses on one of the topics
studied in MA593. MA593 is being withdrawn. No consultation with other schools is
necessary.
6.
The level of the module (e.g. Certificate [C], Intermediate [I], Honours [H] or
Postgraduate [M]): MA609 - H-Level; MA969 - M-Level
7.
The number of credits and the ECTS value which the module represents: 15 (ECTS
7.5)
8.
Which term(s) the module is to be taught in (or other teaching pattern): Autumn or
Spring term
9.
Prerequisite and co-requisite modules:
Prerequisite modules: MA321 (Calculus and Mathematical Modelling), MA322 (Proofs and
Numbers), MA323 (Matrices and Probability) at Stage 1 and MA552 (Analysis), MA553
(Linear Algebra), MA588 (Mathematical Techniques and Differential Equations) at Stage 2,
or students must have studied material equivalent to that covered in these modules.
There are no co-requisite modules.
10. The programmes of study to which the module contributes:
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For the H-level module: BSc (Hons) Mathematics (including programme with a year in
industry), BSc (Hons) Mathematics with a Foundation Year, MMath Mathematics, Graduate
Diploma in Mathematics
For the M-level module: International MSc in Mathematics and its Applications, MSc in
Mathematics and its Applications, MMath Mathematics
For the MMath Mathematics and MMathStat Mathematics and Statistics programmes,
either the H-level or the M-level module may be taken, but not both.
11. The intended subject specific learning outcomes
On successful completion of this module, H-level students will:
a) understand basic geometric objects such as curves and surfaces;
b) be able to construct and manipulate the Frenet frames for plane and space curves;
c) be able to analyse surfaces in three-dimensional space by calculating various
quantities, e.g., the first and second fundamental forms, Gauss curvatures and mean
curvatures;
d) gain an understanding of basic geometric concepts such as geodesics and minimal
surfaces.
On successful completion of this module, M-level students will also:
e) have developed an awareness of the link between classical geometry and modern
applications;
f) be able to apply their geometric understanding to specific examples at the forefront
of the discipline
g) be able to evaluate current research critically.
12. The intended generic learning outcomes
On successful completion of this module, H-level students will have:
a) matured in their problem formulating and solving skills;
b) an enhanced ability to communicate mathematical statements;
c) enhanced skills to associate abstract geometrical concepts with concrete examples;
d) consolidated their grasp of a wide variety of mathematical methods and skills.
On successful completion of this module, M-level students will also have:
e) an enhanced ability to reason and deduce confidently from given definitions and
constructions;
f) an enhanced ability to read independently and manage their time.
13. A synopsis of the curriculum
The main aim is to give an introduction to the basics of differential geometry, keeping in
mind the recent application in mathematical physics and the analysis of pattern recognition.
The synopsis may include:
(a) Theory of curves: Regular plane and space curves. Tangent vectors. Arclength
parameterisation. Curvature and Euclidean invariants. The Frenet formula.
(b) Geometry of surfaces: Regular parameterised surface. The tangent plane.
Curvature of a curve on a surface. First and second fundament forms. Shape
operator. Gaussian curvature and mean curvature.
(c) Geodesics and Minimal surfaces: The Christoffel symbols. Geodesics. The EulerLagrange equations. The Gauss-Bonnet Theorem. Minimal surfaces.
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Possible other topics may include: Evolution of curves and surfaces as integrable
systems: Invariant curve evolution. The mean curvature flow. Riemannian metrics,
connections, curvatures and geodesics.
In addition, for M-level students, the connection with integrable systems; curves evolution in
Riemannian manifolds with constant curvature and Moving frames.
14. Indicative Reading List
V A Toponogov, “Differential Geometry of Curves and Surfaces A Concise Guide’’,
Birkhäuser Boston, 2006 (E-book available in the Templeman Library).
M P Carmo, “Differential Geometry of Curves and Surfaces A Concise Guide’’,
Prentice-Hall, 1976.
V Y Rovenskii, “Geometry of Curves and Surfaces with MAPLE’’, Birkhäuser Boston,
2000.
Extra reading list for M-Level students:
I A Taimanov, “Lectures on Differential Geometry’’, EMS series of lectures in
Mathematics, 2008.
C Rogers and W K Schief, “Bäcklund and Darboux transformations : geometry and
modern applications in soliton theory’’, Cambridge University Press, 2002.
15. Learning and Teaching Methods, including the nature and number of contact hours
and the total study hours which will be expected of students, and how these relate to
achievement of the intended module learning outcomes
Number of contact hours: 42-48 hours, including approx. 36 lectures and 6-12 example
classes (1-hour)
Number of independent learning hours: 102-108 hours.
Total study hours: 150.
Contact hours comprise a mix of lectures and examples classes. Student study hours will
be distributed between consolidation of lecture material, the working of exercises on
exercise sheets and assessed exercises, checking their work and exam preparation.
Lectures and example classes address learning outcomes 11(a)-(d) and 12(a)-(d).
In addition, for the M-level students, example classes cover learning outcomes 11(e),(f) and
12(e) and directed reading addresses learning outcomes 11(e)-(g) and 12(e),(f).
16. Assessment methods and how these relate to testing achievement of the intended
module learning outcomes
The module will be assessed by examination (80%) and coursework (20%).
Coursework: This will normally consist of open-book assignments of unseen problems,
with different assessments for H- and M-level students. The coursework will assess
learning outcomes 11(a)-(d) and 12(a)-(d). In addition, for M-level students, the
coursework will assess learning outcomes 11(e)-(g) and 12(e),(f).
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Examination: This will be a 3-hour written examination in the Summer term that
consists of multi-part questions requiring a mix of short and long answers that test
at varying levels of proficiency learning outcomes 11(a)-(d) and 12(a)-(d). The
examination for M-level students will assess their level of understanding to an
enhanced level, additionally assessing learning outcomes 11(e),(f) and 12(e).
17. Implications for learning resources, including staff, library, IT and space: None-this is
an existing module
18. The School recognises and has embedded the expectations of current disability
equality legislation, and supports students with a declared disability or special
educational need in its teaching. Within this module we will make reasonable
adjustments wherever necessary, including additional or substitute materials,
teaching modes or assessment methods for students who have declared and
discussed their learning support needs. Arrangements for students with declared
disabilities will be made on an individual basis, in consultation with the University’s
disability/dyslexia support service, and specialist support will be provided where
needed.
19. Campus(es) or Centre(s) where module will be delivered: Canterbury
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