好色tv

- Academic Registry


MA30347: Differential geometry of curves and surfaces

[Page last updated: 22 May 2025]

Academic Year: 2025/26
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 Summary: EXCB 100%
Assessment Detail:
  • MA32047 Closed-book written examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA12004 OR take MA12011 OR take MA12013 OR take MA10207 OR take MA10274
In taking this module you cannot take MA30039
Learning Outcomes: By the end of this unit, you will be able to:
  • Apply the methods of calculus to geometrical problems.
  • Compute the curvatures of curves and surfaces and demonstrate an understanding of the geometric significance of these quantities.



Synopsis: Differential geometry is the use of calculus to study geometric objects, and some of the earliest applications of calculus were in fact to study curves and surfaces. You will combine tools from algebra and analysis to describe some of the milestones from work of Euler, Gauss and others in the 18th and 19th centuries, such as the "Theorema Egregium" that the Gauss curvature of a surface depends only on distances measured on the surface and not on how the surface is embedded in space.

Content: Euclidean motions and congruence. Curvature of curves. Torsion of Frenet curves. Frenet equations and Frenet Theorem. First fundamental form and intrinsic properties. Shape operator and second fundamental form. Gauss, mean and principal curvatures. Theorema Egregium. Euler's Theorem. Additional topics from: Turning number and Theorem of Turning Tangents. Four Vertex Theorem. Fenchel's Theorem. Parallel transport. Conformal parametrisations. Variation of area and minimal surfaces. Ruled and developable surfaces. Geodesic and normal curvature of curves and surfaces. Covariant derivatives. Geodesic equation. Geodesic polar coordinates, Gauss Lemma. Gauss equation. Minding's Theorem. Gauss-Codazzi equations and Fundamental Theorem of Surfaces. Gauss-Bonnet Theorem.

Course availability:

MA30347 is Optional on the following courses:

Department of Mathematical Sciences
  • 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 5)
Department of Physics
  • 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 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
  • 好色tv 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.
  • Find out more about these and other important University terms and conditions here.