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# MMath Mathematics and Statistics / Course details

Year of entry: 2021

## 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, 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.

### 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 orientation of bases in vector space
• State the Euler Theorem about rotations in E3. Calculate the axis and an angle of rotation in E3 for an orthogonal operator preserving orientation.
• Define a differential 1-form in En. 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.

### Syllabus

1              Scalar produce and orthonormal bases in IRn.  Affine and Euclidean point spaces. Orientation.  Vector product in IE3..  Geometric meaning of determinant of linear operator.  Isometries of IE2 and IE3.  The Euler theorem

2              Differential forms onIE2 and E3.  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 IE3 and quadratic curves on conic sections

5              Elements of projective geometry.  Projective line IRP4, projective plane PR2.  Projective transformations.  Projective cross-ratio as projective invariant classification of quadratics

### 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 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Benjamin Smith Unit coordinator