Undergraduate Course: Differentiable Manifolds (MATH10088)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||This course is an introduction to differentiable manifolds from an intrinsic point of view, leading to classical theorems such as the generalised Stokes' theorem. It extends the subject matter of Y3 Geometry from surfaces (embedded in R^3) to differentiable manifolds of arbitrary dimension (not necessarily embedded in another space). This provides the necessary concepts to start studying more advanced areas of geometry, topology, analysis and mathematical physics.
The course will include the following topics:
- Smooth manifolds, the manifold topology and submanifolds as level sets.
- Tangent and cotangent spaces, derivative of a smooth map.
- Tangent bundle, vector fields, derivations, flows, Lie derivative.
- Vector bundles, tensor fields.
- Differential forms, Cartan calculus, de Rham complex.
- Orientation, integration, Stokes's theorem.
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling
|High Demand Course?
Course Delivery Information
|Academic year 2020/21, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 7,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S1 (December)||2:00|
On completion of this course, the student will be able to:
- Explain the concept of a manifold and give examples.
- Perform coordinate-based and coordinate-free calculations on manifolds.
- Describe vector fields from different points of view and indicate the links between them.
- Work effectively with tensor fields and differential forms on manifolds.
- State and use Stokes' theorem.
|Recommended in addition to materials provided:|
(*) John Lee, Introduction to smooth manifolds, Springer 2012
Michael Spivak, Calculus on manifolds, Benjamin, 1965
Theodor Broecker & Klaus Jaenich, Introduction to Differential Topology, CUP 1982
(*) Loring Tu, Introduction to Manifolds, Springer 2010
(*) are available to download from the University Library
|Graduate Attributes and Skills
|Course organiser||Prof Josť Figueroa-O'Farrill
Tel: (0131 6)50 5066
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045