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 | |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
This subject lies in the overlap between pure and applied mathematics; it is related to topology on the one hand, and differential equations and vector calculus on the other hand. The unit will discuss the main concepts and theorems of the subject, and the techniques which it provides for calculations in applied mathematics and mathematical physics. The unit will provide useful background for level 4 units in General Relativity, Algebraic Topology, and units under consideration for introduction in the future on Dynamical Systems and Lie Groups. It will include both conceptual results and methods of calculation, so there is material to suit students interested in pur mathematics and also those interested in mathematical techniques.
Aims
To introduce the main tools of the theory of differentiable manifolds.
Syllabus
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.
At the end of the unit students should:
Have facility with the algebra and calculus of differential forms, including the wedge product and exterior derivative
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, problem sheets.
The assessment mark for Differentiable Manifolds 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.
The books by Spivak and Arnold are more advanced. Spivak in particular is a good comprehensive reference.