Unit name | Differentiable Manifolds 4 |
---|---|
Unit code | MATHM2900 |
Credit points | 20 |
Level of study | M/7 |
Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
Unit director | Professor. Robbins |
Open unit status | Not open |
Pre-requisites | |
Co-requisites | |
School/department | School of Mathematics |
Faculty | Faculty of Science |
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.
Aims
To introduce the main tools of the theory of differentiable manifolds.
Syllabus
Relation to Other Units
This unit is a more advanced version of the Level 3 unit, Differentiable Manifolds 3. The lectures for Differentiable Manifolds 3 and Differentiable Manifolds 34 are the same, but the problem sheets and examination questions for Differentiable Manifolds 34 are more challenging. Students may NOT take both Differentiable Manifolds 3 and Differentiable Manifolds 34.
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.
At the end of the unit students should:
Transferable Skills:
Mathematical skills: Knowledge of differentiable manifolds; facility with differential forms, tensor calculus, connections; geometrical reasoning
General skills: Problem solving and logical analysis; Assimilation and use of complex and novel ideas.
Lectures, problem sheets.
The assessment mark for Differentiable Manifolds 34 is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.
Note that the examination for Differentiable Manifolds 34 is different from the examination for Differentiable Manifolds 3, and will be of a standard appropriate to a Level M unit.
The books by Spivak and Arnold are more advanced. Spivak in particular is a good comprehensive reference.