![]() | Department of Mathematics & Computer Science | |||
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Credits: 20 | Convenor: Dr. J. Hunton | Semester: 1 |
Prerequisites: | essential: MC240, MC241, MC224 | |
Assessment: | Individual and group coursework: 15% | Three hour examination in January: 85% |
Lectures: | 32 | Problem Classes: | 10 |
Tutorials: | none | Private Study: | 104 |
Labs: | none | Seminars: | 4 |
Project: | none | Other: | none |
Surgeries: | none | Total: | 150 |
The Earth is round, but maps are drawn on flat pieces of paper. When the map is only a street plan of a city this doesn't matter much, but when the map is of a whole country there can be big discrepancies between the real geography and what the map appears to describe. In a similar manner, Einstein's theory of General Relativity tells us that the Universe is curved, but it's much easier to do the mathematics describing processes in physics as if, at least in local regions, the Universe was flat. The questions of how can we reconcile our ability to do mathematics (drawing curves, measuring distances, differentiating functions and so on) on flat surfaces and spaces with our need to deal with undulating land and curved space-time form the basis of Differential Geometry.
This is an introductory course and will concentrate on presenting some of the ideas of the subject through a study of low dimensional questions and simple geometric objects. It will begin with the study of the most simple geometric examples, namely curves in the plane, and will then proceed to elaborate the ideas needed for studying those to examine more complicated geometric entities. We will see that at each stage we can abstract and generalise the concepts we used before to the next level of complexity, thus the module ought to `reinforce' itself as we go along: the same ideas will keep cropping up again and again.
There will be a group project which will involve applying the concepts of the module to the mathematics of map-making (cartography).
To know the definitions of and understand the key concepts introduced in this module.
To be able to understand, reproduce and apply the main results and proofs in this module.
To be able to solve routine problems in the subject matter covered.
To be aware of some of the applications of the subject.
To give the opportunity for participation in a group project and presentation.
The ability to present arguments and solutions in a coherent and logical form.
The ability to use the techniques taught within the module to solve problems.
Improved ability in group work and presentation skills.
After about three or four introductory lectures explaining the nature of the module and the subject matter, the module splits into four main sections.
§1 Smooth curves in
§2 Smooth curves in
§3 Surfaces in
§4 Surfaces not in
Group Project. Elementary mathematical cartography.
There will be a series of pieces of work set during the semester which together will count for 10% of the final mark. There will be a short group project on aspects of the module run part way through the semester which will count for 5% of the final mark.
There will be six questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to four questions. All questions will carry equal weight. The examination will be worth 85% of the final mark.
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Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
MCS Web Maintainer
This document has been approved by the Head of Department.
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