Module MATH3021: Differential Geometry III

Department: Mathematical Sciences

MATH3021: Differential Geometry III

Prerequisites

• Mathematical Methods II (MATH2811) OR Analysis in Many Variables (MATH2031) AND Analysis I (MATH1051).

Corequisites

• None.

Excluded Combination of Modules

• None.

Aims

• To provide a basic introduction to the theory of curves and surfaces, mostly in 3-dimensional Euclidean space.
• The essence of the module is the understanding of differential geometric ideas using a selection of carefully chosen interesting examples.

Content

• Curves.
• Surfaces in n-dimensional real space.
• First Fundamental Form.
• Mappings of surfaces.
• Geometry of the Gauss map.
• Intrinsic metric properties.
• Theorema Egregium.
• Geodesics.
• Minimal surfaces.
• Gauss-Bonnet Theorem.

Learning Outcomes

• By the end of the module students will:
• Be able to solve seen and unseen problems on given topics.
• Be able to reproduce theoretical mathematics in the field of Differential Geometry.
• Have a knowledge and understanding of fundamental theories and abstract concepts of this field demonstrated through one or more of the following topic areas: curves and surfaces in Euclidean space, first and second fundamental form, mappings of surfaces, geometry of the Gauss map, Gaussian and mean curvature, intrinsic metric properties of surfaces (the Theorema Egregium), curves of shortest length on a surface (geodesics), Gauss-Bonnet theorem.

• Students will have highly specialised and advanced mathematical skills which will be used with minimal guidance in the following areas: Differential Geometry.
• In addition, students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: abstract reasoning.

• Students will develop mathematical skills in the following areas: abstract reasoning, problem solving, spatial awareness.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problems classes show how to solve example problems in an ideal way, also revealing the thought processes behind such solutions.
• Formative assignments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four assignments to be submitted.

■ Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.