Study information

# Applied Differential Geometry - 2024 entry

MODULE TITLE CREDIT VALUE Applied Differential Geometry 15 MTH3013 Dr Hamid Alemi Ardakani (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 25
DESCRIPTION - summary of the module content
On this module, you will have the opportunity to study mathematical topics involving differential geometry of curves and surfaces, and calculus on manifolds. You will learn about various topics in differential geometry such as curves in space and curvature, the Frenet–Serret equations, manifolds and coordinate charts, classification of surfaces, the fundamental equations of surfaces such as the Gauss and Codazzi-Mainardi equations, the Gaussian, mean and principal curvatures of a surface patch, and the Gauss-Bonnet theorem. You become familiar with differential forms, integration and differentiation of differential forms, and the generalised Stokes’ theorem.

Pre-requisite modules: MTH2004 Vector Calculus and Applications, MTH2003 Differential Equations. Co-requisite: MTH2011 Linear Algebra.
AIMS - intentions of the module

The module aims to develop your knowledge of differential geometry of curves and surfaces. By taking it, you will gain a better understanding of manifolds, their mathematical description, and calculus on manifolds. Furthermore, the module introduces the theory differential forms from a geometric viewpoint. By learning advanced topics in differential geometry, the module aims to provide a solid foundation for intrinsic differential geometry and the theory of General Relativity.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. Demonstrate a working knowledge of the mathematical representation of curves and surfaces.
2. Calculate curvature, prove and verify the local and global versions of the Gauss–Bonnet theorem.
3. Demonstrate a working knowledge of differential forms, the generalised Stokes’ theorem, calculus on manifolds, and geodesics

Discipline Specific Skills and Knowledge

4. Reveal an understanding of the fundamental concepts of the differential geometry of curves and surfaces, differential forms and tensor calculus, and appreciate their relevance to many areas of mathematics.

Personal and Key Transferable / Employment Skills and Knowledge

5. Display enhanced theoretical and analytical skills in geometry and advanced calculus, and show competence in modelling geometric objects in computer graphics.
6. Display enhanced problem-solving skills.
7. Demonstrate self-management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module
Topics will include:

Geometry of Curves in Space: curvature, torsion, the Frenet–Serret frame, osculating plane and osculating sphere.

Manifolds and Geometry of Surfaces: manifolds and coordinate charts, transformation of coordinates, parameterised surfaces, the first and second fundamental forms, the Gaussian and mean curvatures, minimal surfaces, Gauss’ equations and the Christoffel symbols, the Weingarten map, the Codazzi–Mainardi equations, Gauss’ Theorema Egregium, the Gauss–Bonnet theorem and geometry of geodesics.

Differential Forms: families of forms, wedge product, exterior derivative, pullbacks, integration of forms, the Hodge star operator, the Laplace–Beltrami operator, the generalised Stokes’ theorem, and the Gauss-Bonnet theorem from differential forms perspective.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures (33 hours) Guided Independent Study 117 Coursework preparation; private study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Un-assessed coursework is assigned to the students, and a sketch of solutions to these are provided. 3 hours per week 1-7 Written and verbal feedback is provided during lectures and office hours.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1 – assessed problem sheet 10 10 hours 1-7 Annotated script and written/verbal feedback
Coursework 2 – assessed problem sheet
10 10 hours 1-7 Annotated script and written/verbal feedback
Written Exam 80 2 hours 1-7 On request

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Coursework 1 – assessed problem sheet Coursework 1 1-7 Referral/deferral period
Coursework 2 – assessed problem sheet Coursework 2 1-7 Referral/deferral period
Written Exam Written Exam 1-7 Referral/deferral period

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener