MATH3021
Lecturer : Norbert Peyerimhoff
Term : Michaelmas 2012/13
Lectures :Problem Classes :
Literature
The following is a list of books on which the lecture is based. They are available in the library. Although we will not follow a books strictly, the material can be found in them and they may sometimes offer a different approach to the material.Assignments
|
Homework |
Date | Hand in | Solutions |
| Questions 2, 6 (i) (ii), 10 (i) (ii) (iii) | 24.10.2012 | 1.11.2012 | Solutions pdf |
| Questions 17, 19 (i) (ii), (iii), 29, 36 | 29.11.2012 | 12.12.2012 | Solutions pdf |
Content of Lectures
| Date | Content |
| Wednesday, 10 October 2012 (Week 1) | Motivation (curves, surfaces and aims of the course), notions of curves (smooth, trace, tangent vector, regular, unit tangent vector, unit speed, singular), examples |
| Thursday, 11 October 2012 (Week 1) | parameter change, arc-length, arc-length parametrisation, examples |
| Wednesday, 17 October 2012 (Week 2) | Curvature of a space curve, tangent vectors, normal vectors and curvature of a unit speed plane curve, examples, curvature for plane curves with arbitrary parametrisation, examples |
| Thursday, 18 October 2012 (Week 2) | Vertices and inflection points of plane curves, examples, 4 vertex Theorem, Fundamental Theorem of local theory of plane curves |
| Wednesday, 24 October 2012 (Week 3) | Radius and centre of curvature, evolute, examples, involute, example, curvature of a space curve |
| Thursday, 25 October 2012 (Week 3) | unit tangent, unit normal and binormal vector of a curve, osculating plane, moving frame, Serret-Frenet equations, curvature and torsion for non unit speed space curves |
| Wednesday, 31 October 2012 (Week 4) | continuation: curvature and torsion for non unit speed space curves, geometric meaning of curvature and torsion, vanishing torsion for space curves in a plane |
| Thursday, 1 November 2012 (Week 4) | fundamental theorem of local theory of space curves, Taylor expansion of space curves up to order 3 in terms of curvature and torsion, open sets in R^n, examples of smooth functions R^n -> R^m, definition of Jacobi matrix and derivative |
| Wednesday, 7 November 2012 (Week 5) | Examples of Jacobi matrices and derivatives, implicit function theorem, example, definition of a surface, local parametrisation and coordinate chart |
| Thursday, 8 November 2012 (Week 5) | Graphs as surfaces, examples, example of sphere as being parametrised with 6 local parametrisations, example of sphere as preimage of a function |
| Wednesday, 14 November 2012 (Week 6) | Regular values and critical points, surfaces as preimages of regular values, examples, definition of a diffeomorphism |
| Thursday, 15 November 2012 (Week 6) | Change of parametrisations, surfaces of revolution, parallels, meridians, canal surfaces, ruled surfaces, examples |
| Wednesday, 21 November 2012 (Week 7) | One sheeted hyperboloid as doubly ruled surface, tangent vectors and tangent plane |
| Thursday, 22 November 2012 (Week 7) | Examples of surfaces and their coordinate tangent vectors, tangent plane of an implicitly defined surface, first fundamental form (FFF), coefficient functions E,F,G of first fundamental form in a local parametrisation |
| Wednesday, 28 November 2012 (Week 8) | Coefficients of FFF for surface of revolution, arc-length of curves in surfaces, upper half-space model of the hyperbolic plane, angle between coordinate curves |
| Thursday, 29 November 2012 (Week 8) | Area of a region in a surface, examples (zone of a sphere, torus of revolution) |
| Wednesday, 5 December 2012 (Week 9) | Further examples of areas (hyperbolic plane, helicoid), definition of a smooth map between surfaces, independence of this definition from the local parametrisation, definition of Gauss map, Gauss map for local parametrisations |
| Thursday, 6 December 2012 (Week 9) | Gauss map and the orientability problem, examples of Gauss maps, the Moebius strip and non-orientability |
| Wednesday, 12 December 2012 (Week 10) | Derivative of a smooth map as a linear map, independence of the derivative of the local parametrisation, examples itself |
| Thursday, 13 December 2012 (Week 10) | Derivative of the Gauss map as a linear map from the tangent space of a surface to itself, definition of a local and a global isometry, example |