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  Differential geometry is the study of geometry using methods of calculus and linear algebra. It has numerous applications in science and mathematics.
This course is an introduction to this classical subject in the context of curves and surfaces in euclidean space. 
Course description 
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 classic 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. One of the main aims will be to quantify the notions of curvature and shape of surfaces. The culmination of the course will be the introduction of the hyperbolic plane, and a sketch proof of the GaussBonnet theorem, a profound result which relates the curvature of surfaces to their topology.
Syllabus:
Curves in Euclidean space: regularity, velocity, arclength, FrenetSerret frame, curvature and torsion, equivalence problem.
Calculus in R^n: tangent vectors, vector fields, differential forms, moving frames, connection forms, structure equations.
Surfaces in Euclidean space: regularity, first and second fundamental forms, curvatures (principal, mean, Gauss), isometry, Gauss's Theorema Egregium, geodesics on surfaces, the hyperbolic plane, integration of forms, statement of Stokes' theorem, Euler characteristic, GaussBonnet theorem (sketch proof).

Information for Visiting Students
Prerequisites  Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.

High Demand Course? 
Yes 
Course Delivery Information

Academic year 2024/25, 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
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)  Geometry (MATH10074)  2:120  
Learning Outcomes
On completion of this course, the student will be able to:
 State definitions and theorems and present standard proofs accurately without access to notes/books.
 Compute the FrenetSerret frame of space curves and determine their torsion and curvature in simple examples.
 Work accurately with differential forms and perform the basic operations of wedge product and exterior derivative.
 Compute the first and second fundamental forms and curvatures of a surface in simple examples.
 Apply theory developed in the course to solve simple problems.

Reading List
Recommended books for supplementary reading (not essential):
Differential forms and Applications, Manfredo P. Do Carmo, Springer 1994
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo, PrenticeHall 1976

Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  Geom 
Contacts
Course organiser  Dr Johan Martens
Tel: (0131 6)51 7759
Email: Johan.Martens@ed.ac.uk 
Course secretary  Miss Greta Mazelyte
Tel:
Email: greta.mazelyte@ed.ac.uk 

