Unit name | Differentiable Manifolds |
---|---|

Unit code | MATH32900 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |

Unit director | Professor. Robbins |

Open unit status | Not open |

Pre-requisites |
MATH20100 Ordinary Differential Equations 2 plus either MATH 20901 Multivariable Calculus (or equivalently Calculus 2) or MATH20200 Analysis 2. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

To introduce the main tools of the theory of differentiable manifolds.

General Description of the Unit

A differentiable manifold is a space which looks locally like Euclidean space but which globally may not. Familiar examples include spheres, tori, regular level sets of functions f(x) on Rn , the group of invertible n x n matrices.

In the unit we develop the theory of vector fields, flows and differential forms mainly for Rn but with a view towards manifolds, in particular surfaces in R3.

The theory of differentiable manifolds extends ideas of calculus and analysis on Rn to these non-Euclidean spaces. An extensive subject in its own right, the theory is also basic to many areas of mathematics (eg, differential geometry, Lie groups, differential topology, algebraic geometry) and theoretical physics and applied mathematics (eg, general relativity, string theory, dynamical systems). It is one of the cornerstones of modern mathematical science.

Important elements in the theory are i)vector fields and flows, which provide a geometrical framework for systems of ordinary equations and generalise notions of linear algebra, and ii) differential forms and the exterior derivative. Differential forms generalise the line, area and volume elements of vector calculus, while the exterior derivative generalises the operations of grad, curl and divergence. The calculus of differential forms generalises and unifies a number of basic results (eg, multidimensional generalisations of the fundamental theorem of calculus: Green's theorem, Stokes' theorem, Gauss's theorem) whilst at the same time bringing to light their geometrical aspect.

Relation to Other Units

The material on Stokes' theorem is relevant to simplicial homology, which is treated in the Level 4 unit Algebraic Topology from a different point of view. The unit complements material in Topics in Modern Geometry 3 (MATH30001, 10cp) and Lie groups, Lie algebras and their representations (MATHM0012, 10cp).

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Learning Objectives

At the end of the unit students should:

- Know and understand the definition of vector fields and flows; be able to calculate flows for simple examples.
- Know and understand the definition of the Jacobi bracket, be able to derive its properties and compute it in examples.
- Know and understand Frobenius integrability theorem and its proof, and be able to apply it to systems for first order PDE's
- Have facility with the algebra and calculus of differential forms, including the wedge product and exterior derivative
- Know and understand the Poincaré lemma and its proof, and be able to apply it
- Know and understand Stokes' theorem for singular cells and its proof; be able to apply it; be familiar with its extension to manifolds.
- Be able to apply the material in the unit to unseen situations

Transferable Skills

Mathematical skills: Knowledge of differentiable manifolds; facility with differential forms, tensor calculus; geometrical reasoning

General skills: Problem solving and logical analysis; Assimilation and use of complex and novel ideas

Lectures, lecture notes, problem sheets, solutions. Informal problems classes.

100 Examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Typeset lecture notes will be provided. There is no single required or recommended text. The following are standard texts that cover some or all of the material. More information on these references is provided in the lecture notes.

- JH and BB Hubbard, "Vector calculus, linear algebra and differential forms: A unified approach", 2 ed, Prentice Hall
- B Schutz, Geometrical methods in mathematical physics, Cambridge University Press
- W Darling, Differential forms and Connections, Cambridge University Press
- M Spivak, A comprehensive introduction to differential geometry, vol 1, Publish or Perish, Berkeley.
- V Arnold, Mathematical methods of classical mechanics, Springer-Verlag.