BSc Computer Science and Mathematics

Year of entry: 2023

Course unit details:Introduction to Geometry

Unit code MATH20222 10 Level 2 Semester 2 Department of Mathematics No

Overview

This course unit introduces the basic ideas of Euclidean and affine geometry, differential forms, conic sections and the first ideas of projective geometry. These notions permeate much of modern mathematics and its applications.

Aims

To give an introduction to the basic ideas of geometry and topology.

Learning outcomes

On successful completion of this module students will be able to:

• Calculate basic algebraic properties of linear operators and Euclidean vector spaces and describe their geometric intuitions.
• Classify how orthogonal operators act in two and three-dimensional Euclidean vector spaces.
• Apply tools in differential geometry, such as differential forms and vector fields, to analyze curves in affine space.
• Classify conic sections and calculate their geometric invariants.
• Describe projective space and apply affine and projective transformations.

Syllabus

1. Euclidean vector spaces

• Matrix representations of linear operators and vectors
• Scalar and vector products, and the geometric interpretation of the determinant
• Orientation of bases and linear operators
• Euler's Rotation Theorem

2. Introduction to Differential Geometry

• Affine space and curves
• Differential 1-forms and their integrals over curves

3. Affine geometry

• Geometric and algebraic definitions of conic sections
• Affine transformations

4. Projective geometry

• Introduction to projective space
• Projective transformations
• Applications to conic sections

Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework; Weighting within unit 20%
• End of semester examination; Weighting within unit 80%

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.

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;

Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

Teaching staff

Staff member Role
Benjamin Smith Unit coordinator
Simon Peacock Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page