MA30039: Differential geometry of curves & surfaces
[Page last updated: 15 October 2020]
Academic Year:  2020/1 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Honours (FHEQ level 6) 
Period: 

Assessment Summary:  EX 100% 
Assessment Detail: 

Supplementary Assessment: 

Requisites:  Before taking this module you must take MA20216 AND take MA20218 AND take MA20219 
Description:  Aims: This course will use vector calculus to develop the local differential geometry of curves and surfaces in Euclidean 3space. In this way, an introduction is given to an area of mathematics which has been the subject of active research for over 200 years. 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. Content: Topics will be chosen from the following: * Parametrizations, tangent spaces, tangent maps. * Euclidean motions. * Curves: length of curves; arclength; normal fields; curvatures and torsion; normal connection; parallel transport; Frenet curves; Frenet formulae; fundamental theorem; isoperimetric inequality; fourvertex theorem. * Surfaces: induced metric; conformal parametrization; Gauss map; shape operator; mean, Gauss and principal curvatures; curvature line parametrization; covariant derivative/LeviCivita connection; Koszul's formulae; curvature tensor; GaussWeingarten equations; GaussCodazzi 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 surfaces; GaussBonnet theorem. * Special surfaces: minimal surfaces; surfaces of constant mean or Gauss curvature; ruled surfaces; developable surfaces. 
Programme availability: 
MA30039 is Optional on the following programmes:Department of Computer Science

Notes:
