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 many of the following topics: -
- 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
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2018/19, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 5%, Examination 95%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Differentiable Manifolds||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 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.
(*) 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
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
|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