Module Catalogues, Xi'an Jiaotong-Liverpool University   
Module Code: MTH315
Module Title: Geometry of Curves and Surfaces
Module Level: Level 3
Module Credits: 5.00
Academic Year: 2019/20
Semester: SEM1
Originating Department: Mathematical Sciences
Pre-requisites: MTH207
• To introduce the ideas and methods of the classical differential geometry of curves and surfaces in three dimensional Euclidean space.

• To translate intuitive geometrical ideas into mathematical concepts that allow for quantitative study.

• To illustrate geometrical concepts through examples.

This module builds on concepts introduced in vector analysis, and has a wide range of applications in science and technology. It provides a foundation for further study of topics such as Riemannian geometry, continuum mechanics and general relativity.

Learning outcomes 
A Apply techniques from calculus and linear algebra with confidence to the study of geometrical objects

B Define and interpret the meaning of core concepts in the differential geometry of curves and surfaces in 3-dimensional space

C Calculate various quantities (e.g. length, curvature, torsion; fundamental forms, area) for given examples and interpret their geometrical significance

D Discriminate between intrinsic and extrinsic geometrical properties

Method of teaching and learning 
This module will be delivered through a combination of formal lectures and tutorials.
1. Curves in the plane: tangent vectors, arc-length, parametrisation, curvature.

2. Curves in 3-space: curvature and torsion, Frenet-Serret coordinate frame.

3. Geometry of the 2-sphere: great circles, parallel transport, stereographic projection.

4. Surfaces in 3-space: surfaces of rotation, quadrics and other examples, parametrisation, tangent plane, normal vector and orientation.

5. Intrinsic metric quantities: first fundamental form, distance, angle, area.

6. Curvature: the Gauss map, second fundamental form, principal curvatures and vectors, mean and Gaussian curvatures, Gauss’s Theorema Egregium, geodesics.

7. Global properties of surfaces: Euler characteristic and genus, Gauss-Bonnet Theorem.
Delivery Hours  
Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total
Hours/Semester 39    13      98  150 


Sequence Method % of Final Mark
1 Coursework 15.00
2 Final Exam 85.00

Module Catalogue generated from SITS CUT-OFF: 8/20/2019 6:25:49 PM