Applied Differential Geometry - 2023 entry
MODULE TITLE | Applied Differential Geometry | CREDIT VALUE | 15 |
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MODULE CODE | MTH3013 | MODULE CONVENER | Dr Hamid Alemi Ardakani (Coordinator) |
DURATION: TERM | 1 | 2 | 3 |
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DURATION: WEEKS | 11 |
Number of Students Taking Module (anticipated) | 25 |
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The module aims to develop students’ knowledge of differential geometry of curves and surfaces. By taking it, you will gain a better understanding of manifolds, their mathematical description, and calculus on manifolds. Furthermore, the module provides an introduction to differential forms from a geometric viewpoint. By learning advanced topics in tensor calculus, the module aims to provide a solid foundation for the theory of General Relativity.
On successful completion of this module you should be able to:
Module Specific Skills and Knowledge
2. Calculate curvature, prove and verify the local and global versions of the Gauss–Bonnet theorem.
Discipline Specific Skills and Knowledge
Personal and Key Transferable / Employment Skills and Knowledge
6. Display enhanced problem-solving skills.
- Geometry of Curves in Space: curvature, torsion, the Frenet–Serret frame, osculating plane and osculating sphere.
- Manifolds and Geometry of Surfaces: manifolds and coordinate charts, transformation of coordinates, parameterised surfaces, the first and second fundamental forms, Gaussian and mean curvatures, minimal surfaces, Gauss’ equations and the Christoffel symbols, Weingarten and Codazzi equations, the theorem Egregium, the Gauss–Bonnet theorem and geometry of geodesics.
- Differential Forms: families of forms, integrating differential 2-forms, the generalised Stokes’ theorem, the Gauss-Bonnet theorem from differential forms perspective.
- Tensor Algebra and Tensor Calculus: contravariant tensors, covariant and mixed tensors, tensor fields, elementary operations with tensors, partial derivative of a tensor, the Lie derivative, the Riemann tensor, geodesic coordinates, the metric and metric geodesics, the metric connection and the curvature and Weyl tensors, tensor densities, the metric determinant, Stokes’ theorem, and the Euler Lagrange equations and variational methods for geodesics.
Scheduled Learning & Teaching Activities | 33 | Guided Independent Study | 117 | Placement / Study Abroad |
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Category | Hours of study time | Description |
Scheduled Learning and Teaching Activities | 33 |
Lectures (33 hours)
Extensive notes and exercises are provided.
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Guided independent study | 117 | Coursework preparation; private study |
Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
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Un-assessed coursework is assigned to the students, and a sketch of solutions to these are provided. | 3 hours per week | All | Written and verbal feedback is provided during lectures and office hours. |
Coursework | 20 | Written Exams | 80 | Practical Exams |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
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Coursework 1 – assessed problem sheet | 10 | 15 hours | All | Annotated script and written/verbal feedback |
Coursework 2 – assessed problem sheet
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10 | 15 hours | All | Annotated script and written/verbal feedback |
Written Exam – Closed Boo | 80 | 2 hours | All | On request |
Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
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All above | Written Exam (100%) | All | August Ref/Def Period |
Deferrals: Reassessment will be by coursework and/or exam in the deferred element only. For deferred candidates, the module mark will be uncapped.
Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.
information that you are expected to consult. Further guidance will be provided by the Module Convener
Reading list for this module:
Type | Author | Title | Edition | Publisher | Year | ISBN |
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Set | Bachman, D. | A Geometric Approach to Differential Forms | Springer Science & Business Media | 2012 | ||
Set | D'Inverno, R. | Introducing Einstein’s Relativity | Oxford University Press | 1992 | ||
Set | do Carmo, M. P. | Differential Geometry of Curves and Surfaces | Prentice-Hall | 1976 | ||
Set | Pressley, A. D. | Elementary Differential Geometry | Springer | 2010 |
CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | MTH2003, MTH2004 |
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CO-REQUISITE MODULES |
NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Tuesday 17th January 2023 | LAST REVISION DATE | Tuesday 26th September 2023 |
KEY WORDS SEARCH | Differential Geometry, Differential Forms, Tensor Calculus |
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Please note that all modules are subject to change, please get in touch if you have any questions about this module.