Student Records
Programme & Unit Catalogues

## MA30039: Differential geometry of curves & surfaces

Owning Department/School: Department of Mathematical Sciences
Credits: 6      [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Honours (FHEQ level 6)
Period: Semester 1
Assessment: EX 100%
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
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 3-space. 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; 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 surfaces; Gauss-Bonnet 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
• USCM-AFB20 : BSc(Hons) Computer Science and Mathematics (Year 3)
• USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
• USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
• USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 3)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 3)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 3)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
Department of Economics
• UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
• UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
• UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
• UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
• USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
• USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
• USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
• USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
• USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
• USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

 Notes: This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here. Programmes and units are subject to change 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. Undergraduates: Find out more about these and other important University terms and conditions here. Postgraduates: Find out more about these and other important University terms and conditions here.