Bachelor of Science (BSc)

BSc Actuarial Science and Mathematics

  • Duration: 3 years
  • Year of entry: 2025
  • UCAS course code: NG31 / Institution code: M20
  • Key features:
  • Scholarships available
  • Accredited course

Full entry requirementsHow to apply

Fees and funding

Fees

Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £34,500 per annum. For general information please see the undergraduate finance pages.

Policy on additional costs

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Scholarships/sponsorships

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .

Course unit details:
Differential Geometry of Curves and Surfaces

Course unit fact file
Unit code MATH31072
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 2
Available as a free choice unit? No

Overview

This unit will be presented as an introduction to Differential Geometry, where the main focus will be on Curves and Surfaces in two- and three-dimensional Euclidean space. As a mathematical discipline, Differential Geometry explores the geometry of smooth shapes. To do this, it uses tools from various areas of mathematics, such as differential calculus, integral calculus and linear algebra.

In Differential Geometry we use analytic and algebraic techniques to formalise the intuitive notions of distance, angle and curvature to study the extrinsic properties of curves (parameterised by arc length). Some of these properties are then used more generally to study the intrinsic and extrinsic properties of surfaces. For instance, since the time of Euclid it was understood that a straight line provides the shortest distance between two points; however, how do you find this distance if the points lie in the surface of the earth (which is not flat). This leads to the important concept of geodesics where one uses the idea that great circles are locally similar to straight lines in a flat plane.

In this module we will explore several notions around smooth parameterised curves, such as tangent vectors, curvature, osculating planes, and the Frenet moving frame. We will then move to the study of parametric surfaces where we discuss tangent planes, parallel transport, curvatures, geodesics, fundamental forms, Christoffel symbols, and the Gauss-Bonnet theorem.  Many examples will be presented throughout.  

 

Pre/co-requisites

Unit title Unit code Requirement type Description
Analysis and Geometry in Affine Space MATH31061 Pre-Requisite Compulsory
math31072 pre-reqs

Aims

The unit aims to deliver a gentle introduction to fundamental aspects of classical Differential Geometry. It explores and studies the local properties of smooth curves and surfaces by means of differential calculus.

 

Learning outcomes

  • Construct parametric curves in Euclidean space from geometric criteria and define and compute the fundamental notions attached to them such as tangent and normal vectors and arc-length
  • State the fundamental theorem of curve theory and global theorems for closed curves, and deduce simple consequences  
  • Define and analyze the central notions of curvature and torsion of curves, and the Gauss, mean and principal curvatures of surfaces and use them to determine features such as geodesics and parallel transport  
  • Define and calculate moving frames for curves and surfaces and use them to analyze geometric properties
  • Reproduce classical examples of smooth surfaces and compute the first and second fundamental form in some specific instances; deduce properties of surfaces from these  
  • State and explain the Gauss Theorema Egregium and the Gauss-Bonnet Theorem.  
     

Syllabus

  • Parametric Curves (6 lectures, 3 weeks). Velocity, tangent vectors and arc length. Acceleration and curvature. Torsion of space curves. Independence of parameterisation, osculating plane/circle/sphere and idea of n-point contact. Frenet moving frame and Serret-Frenet formula. Evolute and Involute. Examples: conics, helix, and envelopes of families of lines.
  • Theorems for plane curves (4 lectures, 2 weeks).  Rigid motions and the fundamental theorem of curve theory. Crofton’s formula. Total curvature and rotation number.
  • Parametric surfaces (6 lectures, 3 weeks).  Tangent planes (affine and linear). First fundamental form. Curvatures of normal sections. Euler’s theorem and principal curvatures. Gauss and mean curvature. Second fundamental form. Meusnier’s theorem. Lines of curvature, parabolic locus and asymptotic lines. Examples to include spheres, ruled surfaces, developable surfaces, and minimal surfaces.  
  • Geodesics and Parallel Transport (6 lectures, 3 weeks).  Geodesics in terms of curvature. Parallel transport and covariant derivative. Christoffel s

Teaching and learning methods

Teaching in traditional format with 2hrs of lectures and 1hr tutorial per week.  

Assessment methods

Method Weight
Written exam 80%
Written assignment (inc essay) 20%

Feedback methods

Generic feedback made available after exam period

Recommended reading

  • Differential Geometry of Curves and Surfaces. M. do Carmo, 1976.  
  • Differential Geometry: A first course in curves and surfaces. T. Shifrin, 2021.
  • An introduction to Differential Geometry. T. J. Willmore, 1964. cs and Parallel Transport (6 lectures, 3 weeks).  Geodesics in terms of curvature. Parallel transport and covariant derivative. Christoffel symbols. Gauss-Peterson-Codazzi equations. Gauss Theorema Egregium. Gauss-Bonnet Theorem.  

Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

Teaching staff

Staff member Role
Omar Leon Sanchez Unit coordinator
James Montaldi Unit coordinator

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