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MATH3021: DIFFERENTIAL GEOMETRY III

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Type Open
Level 3
Credits 20
Availability Available in 2024/2025
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

  • Analysis in Many Variables II (MATH2031).

Corequisites

  • None.

Excluded Combinations 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

Subject-specific Knowledge:

  • By the end of the module students will: be able to solve novel and/ or complex problems in Differential Geometry.
  • have a systematic and coherent understanding of theoretical mathematics in the field of Differential Geometry.
  • have acquired a coherent body of knowledge of these subjects 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 Theorem Egregium.
  • Curves of shortest length on a surface: geodesics.
  • Gauss-Bonnet theorem.

Subject-specific Skills:

  • In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial Awareness.

Key Skills:

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.
  • Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
  • Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
  • The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 31 Hour42 
Problems Classes8Four in each of terms 1 and 21 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 Hours100 

Formative Assessment

Eight assignments to be submitted.

More information

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