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
Semester: SEM1
Originating Department: Mathematical Sciences
Pre-requisites: MTH207

 Aims • 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 objectsB Define and interpret the meaning of core concepts in the differential geometry of curves and surfaces in 3-dimensional spaceC Calculate various quantities (e.g. length, curvature, torsion; fundamental forms, area) for given examples and interpret their geometrical significanceD 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.
 Syllabus 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

## Assessment

 Sequence Method % of Final Mark 1 Coursework 15.00 2 Final Exam 85.00
 Module Catalogue generated from SITS CUT-OFF: 6/4/2020 7:00:46 AM