Module Details

Module Specification

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.

Module Title

DIFFERENTIAL GEOMETRY

Module Code

MATH349

Year

Session 2023-24

Originating Department

Mathematical Sciences

Faculty

Fac of Science & Engineering

Semester

First Semester

CATS Level

Level 6 FHEQ

CATS Value

Member of staff with responsibility for the module

Dr A Rizzardo

Mathematical Sciences

Alice.Rizzardo@liverpool.ac.uk

10.

Module Moderator

11.

Other Contributing Departments

12.

Other Staff Teaching on this Module

Professor VV Goryunov

Mathematical Sciences

Victor.Goryunov@liverpool.ac.uk

13.

Board of Studies

14.

Mode of Delivery

15.

Location

Main Liverpool City Campus

		Lectures	Seminars	Tutorials	Lab Practicals	Fieldwork Placement	Other	TOTAL
16.	Study Hours			24			36	60
17.	Private Study							90
18.	TOTAL HOURS							150

		Lectures	Seminars	Tutorials	Lab Practicals	Fieldwork Placement	Other
19.	Timetable (if known)

20.	Pre-requisites before taking this module (other modules and/or general educational/academic requirements): MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra

21.	Modules for which this module is a pre-requisite:

22.	Co-requisite modules:

23.	Linked Modules:

24.	Programme(s) (including Year of Study) to which this module is available on a mandatory basis:

25.	Programme(s) (including Year of Study) to which this module is available on a required basis:

26.	Programme(s) (including Year of Study) to which this module is available on an optional basis:

27.

Aims

This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

28.

Learning Outcomes

(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

(S2) Numeracy

29.

Teaching and Learning Strategies

Material is provided in advance of classes for students to study asynchronously. The contact hours consist of one 2-hour active learning session and one 2-hour supported study/drop-in session.

30.

Syllabus

1. Curves in the plane and in space.
2. Surface patches in 3-space. Parametrizations.
3. Distance and the first fundamental form of a surface.
4. Curvature of surfaces and the second fundamental form. Special curves on a surface: principal curves, asymptotic curves, geodesics. Elliptic, hyperbolic and parabolic points.
5. Gauss's theorem on curvature: the intrinsic nature of the Gauss curvature.
6. Geodesics on a surface.
7. The Gauss-Bonnet theorem.

31.

Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Assessment

32.	EXAM	Duration	Timing (Semester)	% of final mark	Resit/resubmission opportunity	Penalty for late submission	Notes
	Final Assessment This is an anonymous assessment.	120		70

33.	CONTINUOUS	Duration	Timing (Semester)	% of final mark	Resit/resubmission opportunity	Penalty for late submission	Notes
	Class Test on campus	60		30