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Unit code: | MATH20222 |
Credit Rating: | 10 |
Unit level: | Level 2 |
Teaching period(s): | Semester 2 |
Offered by | School of Mathematics |
Available as a free choice unit?: | N |
Requisites
NoneAims
To give an introduction to the basic ideas of geometry and topology.
Overview
This course unit introduces the basic ideas of Euclidean and affine geometry, quadric curves and surfaces in Euclidean space, and differential forms, and the first ideas of projective geometry. These notions permeate all modern mathematics and its applications.
Learning outcomes
On successful completion of this module students will be able to:
- Calculate orientation of bases in vector space
- State the Euler Theorem about rotations in E^{3}. Calculate the axis and an angle of rotation in E^{3} for an orthogonal operator preserving orientation.
- Define a differential 1-form in E^{n}. Calculate the values of 1-forms on vectors. Calculate differential of functions and the directional derivative of a function along a vector. Calculate integrals of differential 1-forms over curves.
- Establish relations between analytic and geometric definitions of conic sections. In particular find foci of an ellpse, and find focus and directrix of a parabola given by analytic expressions.
- Find cross-ratio of four collinear points on projective plane. Find projective transformations of conic sections.
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Assessment methods
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; Weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%
Syllabus
1 Scalar produce and orthonormal bases in IR^{n}. Affine and Euclidean point spaces. Orientation. Vector product in IE^{3.. } Geometric meaning of determinant of linear operator. Isometries of IE^{2 }and IE^{3}. The Euler theorem
2 Differential forms onIE^{2} and E^{3}. Exampler: Geometric meaning are of parallelogram, volume of parallelepiped. Integration of differential form over a curve. Exact forms.
3 Conic (quadratic curves) in the plane. Foci of ellipses and byperbolas, Euclidean and affine classification of quadratic curves
4 Cone in IE^{3} and quadratic curves on conic sections
5 Elements of projective geometry. Projective line IRP^{4}, projective plane PR^{2}. Projective transformations. Projective cross-ratio as projective invariant classification of quadratics
Recommended reading
1) David A. Brannan, Geometry, Cambridge University Press, 2011-12-22, 2nd edition.
2) B.A. Dubrovin, A.T. Fomenko, S.P. Novikov. Modern geometry, methods and applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Vol. 93, 1992,
3) Geometry of Differential, forms. Morita (Shigeyuki),AMS,vol.201
4) Barrett O' Neill, Elementary Differential Geometry, Academic Press.
5) Andrew Pressley, Elementary Differential Geometry, Springer;
Feedback methods
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
Study hours
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours
Teaching staff
Hovhannes Khudaverdyan - Unit coordinator