Unit information: Differentiable Manifolds in 2013/14

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 and either MATH20900 or MATH20200
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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

• Vector fields, flows and diffeomorphisms [~5 weeks]: Maps and diffeomorphisms on Rn. ODE's, vector fields, and flows. Jacobi bracket. Frobenius integrability theorem. Parameterized surfaces in R3. Bonnet's theorem.

• Differential forms [~3 weeks]: Tensors. Exterior algebra. Wedge product. Differential forms. Exterior derivative. Poincaré Lemma.

• Stokes' Theorem [~2 weeks]: Integration of forms. Cells and boundaries. Stokes' Theorem. Gauss-Bonnet theorem (nonexaminable).

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.

Intended Learning Outcomes

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

Teaching Information

Lectures, problem sheets.

Assessment Information

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.

Reading and References

• 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.

The books by Spivak and Arnold are more advanced. Spivak in particular is a good comprehensive reference.