Introduction to Geometry
|Unit level:||Level 2|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
To give an introduction to the basic ideas of geometry and topology.
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.
On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in the three-dimensional Euclidean space, and will be acquainted with the ways of generalising these concepts to higher dimensions.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; Weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%
1. Scalar product and orthonormal bases in Rn. Affine and Euclidean point spaces. Orientation. Vector product E3. Affine transformations and isometries. Geometric meaning of determinant of linear operator. Ismoetries of E2 and E3. The Eular theorem.
2. Differentil forms on E2 and E3. Geometric meaning. Examples: area of parallelogram, volume of Parallelepiped. Integral of differential form over a curve. Exact forms.
3. Conics (quadric curves) in the plane. Foci of ellipses and hyperbolas. Euclidean and affine classification or quadric curves.
4. Surfaces in E3. quadric surfaces. Cone in E3 and quadric curves in E2 as conic sections.
5. Elements of projective geometry. Projective line RP1, projective plane RP2. Projective transformations. Cross-ratio as projective invariant. Projective classification of quadrics.
1) Andrew Pressley, Elementary Differential Geometry, Springer;
2) Barrett O' Neill, Elementary Differential Geometry, Academic Press.
3) David A. Brannan, Geometry, Cambridge University Press, 2011-12-22, 2nd edition.
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.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours