 Student Records
Programme & Unit Catalogues

## MA50039: Differential geometry of curves & surfaces

2015/6 Owning Department/School: Department of Mathematical Sciences
6
Masters UG & PG (FHEQ level 7)
Semester 1
CW 25%, EX 75%
• Coursework (CW 25%)
• Examination (EX 75%) Supplementary Assessment: Mandatory extra work (where allowed by programme regulations) Description: Aims:
This course will use methods from multi-dimensional analysis to develop the local differential geometry of curves and surfaces in Euclidean space. In this way, the course provides an introduction to an area of active research in mathematics.

Learning Outcomes:
At the end of the course, the students will be able to apply the methods of calculus with confidence to geometrical problems. They will be able to compute the curvatures of curves and surfaces and understand the geometric significance of these quantities. Through their coursework, the students will be able to demonstrate a deep understanding of an area of differential geometry.

Skills:
Analytic skills T/F A; Problem solving T/F A; Written communication F A.

Content:
Topics will be chosen from the following:
* Parametrization, tangent spaces, tangent maps.
* Euclidean motions.
* Curves: length of curves; arc-length; normal fields; curvatures and torsion; normal connection; parallel transport; Frenet curves, Frenet formulae; fundamental theorem; isoperimetric inequality; four-vertex theorem.
* Surfaces: induced metric; conformal parametrization; Gauss map;shape operator; mean, Gauss and principal curvatures; curvature line parametrization; covariant derivative/Levi-Civita connection; Koszul's formulae; curvature tensor; Gauss-Weingarten equations; Gauss-Codazzi equations; Bonnet's theorem.
* Curves on surfaces: geodesics; geodesic curvature; geodesic polar coordinates; geodesics as local length minimizers; Minding's theorem; Clairaut's theorem; normal curvature; Euler's theorem; Meusnier's theorem; asymptotic lines; curvature lines; Rodrigues' equation; Joachimsthal's theorem; integration on srufaces; Gauss-Bonnet theorem.
* Special surfaces: minimal surfaces; surfaces of constant mean or Gauss curvature; ruled surfaces; developable surfaces. Programme availability:

#### MA50039 is Optional on the following programmes:

Department of Mathematical Sciences

Notes:
* This unit catalogue is applicable for the 2015/16 academic year only. Students continuing their studies into 2016/17 and beyond should not assume that this unit will be available in future years in the format displayed here for 2015/16.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.