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MA3D9-15 Geometry of Curves & Surfaces

Department
Warwick Mathematics Institute
Level
Undergraduate Level 3
Module leader
Weiyi Zhang
Credit value
15
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry
Introductory description

This will be an introduction to some of the ``classical'' theory of differential geometry, as illustrated by the geometry of curves and surfaces lying (mostly) in 3-dimensional space. The manner in which a curve can twist in 3-space is measured by two quantities: its curvature and torsion. The case a surface is rather more subtle. For example, we have two notions of curvature: the gaussian curvature and the mean curvature. The former describes the intrinsic geometry of the surface, whereas the latter describes how it bends in space. The gaussian curvature of a cone is zero, which is why we can make a cone out of a flat piece of paper. The gaussian curvature of a sphere is strictly positive, which is why planar maps of the earth's surface invariably distort distances. One can relate these geometric notions to topology, for example, via the so-called Gauss-Bonnet formula. This is mostly mathematics from the first half of the nineteenth century, seen from a more modern perspective. It eventually leads on to the very general theory of manifolds.

Module web page

Module aims

To gain an understanding of Frenet formulae for curves, the first and second fundamental forms of surfaces in 3-space, parallel transport of vectors and gaussian curvature. To apply this understanding in specific examples.

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

N/A

Learning outcomes

By the end of the module, students should be able to:

  • N/A
Indicative reading list

Books: John McCleary, Geometry from a differential viewpoint Cambridge University Press 1994.
Dirk J. Struik, Lectures on classical differential geometry Addison-Wesley 1950.
M Do Carmo, Differential geometry of curves and surfaces, Prentice Hall.

Subject specific skills

N/A

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.

Study time

Type Required
Lectures 30 sessions of 1 hour (20%)
Tutorials 9 sessions of 1 hour (6%)
Private study 111 hours (74%)
Total 150 hours
Private study description

Review lectured material and work on set exercises.

Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

Students can register for this module without taking any assessment.

Assessment group B1
Weighting Study time
In-person Examination 100%
  • Answerbook Gold (24 page)
Assessment group R
Weighting Study time
In-person Examination - Resit 100%
  • Answerbook Gold (24 page)
Feedback on assessment

Exam feedback.

Past exam papers for MA3D9

Courses

This module is Optional for:

  • Year 1 of TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
  • UCSA-G4G1 Undergraduate Discrete Mathematics
    • Year 3 of G4G1 Discrete Mathematics
    • Year 3 of G4G1 Discrete Mathematics
  • Year 3 of UCSA-G4G3 Undergraduate Discrete Mathematics
  • Year 4 of UCSA-G4G2 Undergraduate Discrete Mathematics with Intercalated Year
  • USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
    • Year 3 of G300 Mathematics, Operational Research, Statistics and Economics
    • Year 4 of G300 Mathematics, Operational Research, Statistics and Economics

This module is Core option list B for:

  • UMAA-GV17 Undergraduate Mathematics and Philosophy
    • Year 3 of GV17 Mathematics and Philosophy
    • Year 3 of GV17 Mathematics and Philosophy
    • Year 3 of GV17 Mathematics and Philosophy
  • Year 3 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Core option list D for:

  • Year 4 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

This module is Option list A for:

  • Year 1 of TMAA-G1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
  • Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
  • Year 1 of TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
  • UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
    • Year 3 of G105 Mathematics (MMath) with Intercalated Year
    • Year 5 of G105 Mathematics (MMath) with Intercalated Year
  • UMAA-G100 Undergraduate Mathematics (BSc)
    • Year 3 of G100 Mathematics
    • Year 3 of G100 Mathematics
    • Year 3 of G100 Mathematics
  • UMAA-G103 Undergraduate Mathematics (MMath)
    • Year 3 of G100 Mathematics
    • Year 3 of G103 Mathematics (MMath)
    • Year 3 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
  • UMAA-G106 Undergraduate Mathematics (MMath) with Study in Europe
    • Year 3 of G106 Mathematics (MMath) with Study in Europe
    • Year 4 of G106 Mathematics (MMath) with Study in Europe
  • Year 3 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
  • UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
    • Year 3 of GF13 Mathematics and Physics
    • Year 3 of GF13 Mathematics and Physics
  • UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
    • Year 3 of FG31 Mathematics and Physics (MMathPhys)
    • Year 3 of FG31 Mathematics and Physics (MMathPhys)
  • Year 4 of UPXA-GF14 Undergraduate Mathematics and Physics (with Intercalated Year)
  • Year 4 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
  • Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year
  • USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
    • Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
    • Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
  • Year 4 of USTA-Y603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)

This module is Option list B for:

  • Year 1 of TMAA-G1PE Master of Advanced Study in Mathematical Sciences
  • Year 3 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
  • Year 4 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
  • USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
    • Year 3 of GG14 Mathematics and Statistics
    • Year 3 of GG14 Mathematics and Statistics
  • Year 4 of USTA-GG17 Undergraduate Mathematics and Statistics (with Intercalated Year)

This module is Option list E for:

  • USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
    • Year 3 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
    • Year 4 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
  • USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
    • Year 3 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
    • Year 4 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
    • Year 5 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)