Undergraduate Course: Differentiable Manifolds (MATH10088)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Credits | 10 |
| Home subject area | Mathematics |
Other subject area | None |
| Course website |
None |
Taught in Gaelic? | No |
| Course description | 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. |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | Yes |
Course Delivery Information
|
| Delivery period: 2014/15 Semester 2, Available to all students (SV1)
|
Learn enabled: Yes |
Quota: None |
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Web Timetable |
Web Timetable |
| Course Start Date |
12/01/2015 |
| Breakdown of Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
98 )
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| Additional Notes |
|
| Breakdown of Assessment Methods (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
|
| No Exam Information |
Summary of Intended Learning Outcomes
- Explain the concept of a manifold and give examples.
- Perform coordinate-based 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.
- Explain the concept of a Riemannian metric |
Assessment Information
| Coursework 5%, Examination 95% |
Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
- Definition of topological manifolds
- Smooth manifolds and smooth maps, partitions of unity
- Submanifolds and implicit function theorem
- Tangent spaces and vector fields from different points of view (derivations, velocities of curves)
- Flows and Lie derivatives
- Tensor fields and differential forms
- Orientation, integration and the generalised Stokes' Theorem
- Basic notions of Riemannian geometry |
| Transferable skills |
Not entered |
| Reading list |
Recommended :
(*) John Lee, Introduction to smooth manifolds, Springer 2012
Michael Spivak, Calculus on manifolds, Benjamin, 1965
Theodor Bröcker & Klaus Jänich, Introduction to Differential Topology, CUP 1982
Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer 1983
(*) Loring Tu, Introduction to Manifolds, Springer 2010
(*) are available to download from the University Library |
| Study Abroad |
Not entered |
| Study Pattern |
Not entered |
| Keywords | DMan |
Contacts
| Course organiser | Dr Pieter Blue
Tel: (0131 6)50 5076
Email: P.Blue@ed.ac.uk |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk |
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