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School of Mathematics

# MATH31061/MATH41016 - 2007/2008

General Information
• Title: Calculus on Manifolds
• Unit code: MATH31061/MATH41061
• Credits: 10 (MATH31061), 15 (MATH41061)
• Prerequisites: Vector calculus
• Co-requisite units: MATH31051 Introduction to Topology
• School responsible: Mathematics
• Member of staff responsible: Dr. Ted Voronov
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## Specification

### Brief Description of the unit

Manifolds are one of the main notions of modern geometry and topology. They are used everywhere in mathematics and its applications. Roughly, manifolds are geometrical objects that can be endowed with coordinates, so that using these coordinates one can apply differential and integral calculus. Thus manifolds are arguably the most natural objects where calculus can be developed. On the other hand, the theory of manifolds provides for calculus and its applications a powerful 'invariant' geometrical language.

### Future topics requiring this course unit

MATH41122 Differential Geometry

### Syllabus

1. Manifolds and smooth maps: examples and definition.
2. Tangent bundle, vectors and tensors. Differential forms as skew-symmetric tensors and their algebra. Manifolds as surfaces in Euclidean space (statement).
3. Commutators of vector fields. Lie derivative. Exterior differential of differential forms. Cartan formula.
4. Topology induced by manifold structure. Compactness and connectedness: recollection and examples. Partition of unity (statement).
5. Oientation. Manifolds with boundary. Integration of differential forms. Stokes Theorem.
6. De Rham cohomology: definition. Examples of non-trivial cohomology classes. Poincare Lemma. De Rham Theorem (statement).

### Teaching and learning methods

Two lectures per week plus one weekly examples class.

### Assessment

Coursework; Weighting within unit 20%
2 hours end of semester examination; Weighting within unit 80%

## Arrangements

Online course materials are available for this unit.