Undergraduate Course: Geometry (MATH10074)

Course Outline
School	School of Mathematics	College	College of Science and Engineering
Credit level (Normal year taken)	SCQF Level 10 (Year 3 Undergraduate)	Availability	Available to all students
SCQF Credits	10	ECTS Credits	5
Summary	This course is an introduction to differential geometry, which is the study of geometry using methods of calculus and linear algebra. The course begins with curves in euclidean space; these have no intrinsic geometry and are fully determined by the way they bend and twist (curvature and torsion). The rest of the course will then develop the theory of surfaces. This will be done in the modern language of differential forms. Surfaces possess a notion of intrinsic geometry and many of the more advanced aspects of differential geometry can be demonstrated in this simpler context. The culmination of the course will be a sketch proof of the Gauss-Bonnet theorem, a remarkable connection between the curvature of surfaces and their topology.
Course description	Curves in Euclidean space, regularity, velocity, arc-length, Frenet-Serret frame, curvature and torsion, equivalence problem. Tangent vectors, vector fields, differential forms, Poincareżs Lemma, moving frames, connection forms, structure equations. Surfaces, first and second fundamental forms, curvature, isometry, Theorema Egregium, geodesics on surfaces, integration of forms, statement of general Stokes' theorem, Euler characteristic, Gauss-Bonnet theorem (sketch proof).

Entry Requirements (not applicable to Visiting Students)
Pre-requisites	Students MUST have passed: Fundamentals of Pure Mathematics (MATH08064) AND Several Variable Calculus and Differential Equations (MATH08063)	Co-requisites
Prohibited Combinations		Other requirements	Students must not have taken MATH10002 Differential Geometry.

Information for Visiting Students
Pre-requisites	None
High Demand Course?	Yes

Course Delivery Information

Academic year 2015/16, Available to all students (SV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 5, Summative Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 69 )
Assessment (Further Info)	Written Exam 95 %, Coursework 5 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework 5%, Examination 95%
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)	MATH10074 Geometry		2:00

Academic year 2015/16, Part-year visiting students only (VV1)		Quota: None
Course Start	Semester 2
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 5, Summative Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 69 )
Assessment (Further Info)	Written Exam 95 %, Coursework 5 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework 5%, Examination 95%
Feedback	Not entered
No Exam Information

Learning Outcomes
1. An ability to compute the Frenet-Serret frame of space curves and determine their torsion and curvature. 2. An ability to perform simple manipulations with differential forms. 3. An understanding of the first and second fundamental forms of a surface and its principal curvatures. An ability to compute simple examples. 4. An ability to calculate geodesics on surfaces in simple examples. 5. An ability to apply Stokes' Theorem to simple examples. 6. An ability to apply the Gauss-Bonnet theorem to simple examples.