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 DIFFERENTIAL GEOMETRY
Code MATH349
Coordinator Dr O Karpenkov
Mathematical Sciences
O.Karpenkov@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ Second Semester 15

Aims

               

This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space.  While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.


Learning Outcomes

1. Knowledge and understanding

After the module, students should have a basic understanding of

a) invariants used to describe the shape of explicitly given curves and surfaces,

b) special curves on surfaces,

c) the difference between extrinsically defined properties and those which depend only on the surface metric,

d) understanding the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

2. Intellectual abilities

< p class="msonormal" xmlns="http://www.w3.org/1999/xhtml"> After the module, students should be able to

a) use differential calculus to discover geometric properties of explicitly given curves and surface,

b) understand the role played by special curves on surfaces.

3. Subject-based practical skills

Students should learn to

a) compute invariants of curves and surfaces,

b) interpret the invariants of curves and surfaces as indicators of their geometrical properties.

4. General transferable skills

Students will improve their ability to

a) think logically about abstract concepts,

b) combine theory with examples in a meaningful way.


Syllabus

1

1.     Curves in the plane and in space.

2.     Surface patches in 3-space. Parametrizations.

3.     Distance and the first fundamental form of a surface.

4.     Curvature of surfaces and the second fundamental form. Special curves on a surface: principal curves, asymptotic curves, geodesics.  Elliptic, hyperbolic and parabolic points.

5.     Gauss''s theorem on curvature: the intrinsic nature of the Gauss curvature.

6.     Geodesics on a surface.

7.     The Gauss-Bonnet theorem.


Recommended Texts

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

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

MATH101; MATH102; MATH103 MATH248 is useful but not required 

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:

Programme:G100 Year:3 Programme:G101 Year:3,4 Programme:G110 Year:3 Programme:G1F7 Year:3 Programme:G1R9 Year:4 Programme:GG13 Year:3 Programme:GN11 Year:3 Programme:GL11 Year:3 Programme:GR11 Year:4 Programme:GV15 Year:3 Programme:BCG0 Year:3 Programme:L000 Year:3 Programme:Y001 Year:3 Programme:MMAS Year:1

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam  2.5 hours  Second  85  Standard University policy    Assessment 2 Notes (applying to all assessments) Class Test Learning outcomes 1a-c, 2a, 3a-b, 4a-d will be assessed in the continuous assessment of practical work. This work is not marked anonymously. Full marks can be obtained for complete answers to FIVE questions. Only the best FIVE answers will be counted. The examination will test learning outcomes 1a-d, 2a, 3a-b, 4a.  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Coursework    Second  15  None: exemption approved November 2007  University policy.  Assessment 1