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Programme & Unit Catalogues

## MA30039: Differential geometry of curves & surfaces

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Honours (FHEQ level 6)
Period: Semester 1
Assessment: EX 100%
Supplementary Assessment: MA30039 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit 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-AFB01 : BSc Computing (Full-time) - Year 3
• USCM-AKB01 : BSc Computing (Full-time with Thick Sandwich Placement) - Year 4
• USCM-AFB20 : BSc (hons) Computer Science and Mathematics (Full-time) - Year 3
• USCM-AKB20 : BSc (hons) Computer Science and Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USCM-AAB20 : BSc (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USCM-AFB13 : BSc (hons) Computer Science with Mathematics (Full-time) - Year 3
• USCM-AKB14 : BSc (hons) Computer Science with Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USCM-AAB14 : BSc (hons) Computer Science with Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 3
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 4
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
• TSMA-APM09 : MSc Mathematical Sciences (Part-time)
Department of Physics
• USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 3
• USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 4
• USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 4

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* 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.