# MST30070 Differential Geometry

This module will offer both a recap of elements of standard Euclidean geometry and give an introduction to the theory of differential geometry. It will, in particular, give examples of non-Euclidean geometries such as spherical geometry etc..

The differential geometry will cover the modern theory of curves and surfaces. Topics covered will be taken from the following list:
CURVES: regular curve, tangent vector, tangent line, reparametrisations, the arc-length function, the Frenet-Serret apparatus and the Frent-Serret theorem for unit speed and non-unit speed curves.
SURFACES: Simple surface, tangent plane, tangent space, co-ordinate transformation and definition of a general surface. Level sets as surfaces. The first and second fundamental forms of a simple surface and brief introduction to Guassian curvature.
(STUDENTS MUST HAVE A LAPTOP COMPUTER.)

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Curricular information is subject to change

Learning Outcomes:

Students must be able to calculate the quantities inherent in the course: tangent vector, tangent line, tangent space, unit-speed reparametrisations etc.. They must be able to state the theorems proved in the course and reproduce some elements of proof. They must demonstrate an understanding of the concepts involved.

More generally students should be able to do the following:

WRITE MATHEMATICS: Students should be able to recognise, read and correctly use standard mathematical symbols and notation, to correctly write a mathematical statement and to recognise when such a statement is not correctly written.

QUESTION: Students should be able to ask pertinent questions themselves, to decide which questions are most relevant, which questions are answerable, which questions they should start with, etc.

UNDERSTAND: Students must be able to understand the reasoning behind any methods or procedures they use and be able to demonstrate that understanding.

PRODUCE EXAMPLES: Students must be able to produce examples themselves, to illustrate a definition, to show a method, to test boundaries of an idea.

Student Effort Hours:
Student Effort Type Hours
Lectures

24

Tutorial

10

Autonomous Student Learning

66

Total

100

Approaches to Teaching and Learning:
lectures, tutorials and problem based learning
Requirements, Exclusions and Recommendations
Learning Requirements:

Students should have a knowledge of introductory analysis (e.g. MST20040) and should have completed a course in Multi-variable calculus at the level of MST20070 or MATH20060.

Module Requisites and Incompatibles
Incompatibles:
MATH40370 - Differential Geometry

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Exam (In-person): IN person final exam n/a Standard conversion grade scale 40% No

70

Quizzes/Short Exercises: Continuous assessment in the form of tutorial quizzes and mid-term exam n/a Standard conversion grade scale 40% No

30

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour