Topic outline

  • General and Announcements

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    Geometry is the field of mathematics concerned with studying the shapes, sizes, and positions of objects. This module provides an introduction to the differential geometry of curves and surfaces. The main focus will be on connecting geometric questions with ideas from calculus and linear algebra, and on using these connections to gain a better understanding of all three subjects.

    The module will cover the following general topics (see the first chapter of the lecture notes for a more detailed summary):

    1. Vector algebra and calculus
    2. The geometry of curves
    3. The geometry of surfaces
    4. Applications (Lagrange multipliers, vector integral theorems)

    The background knowledge needed for the module include calculus (differentiation and integration), as well as some basic vector and matrix algebra. Though there will be some proofs, they will not be a central focus of the module.

    Essential Information

    The Monday and Wednesday, 11:00-12:30 lectures are held at the Fogg Lecture Theatre (Fogg Building).

    Office hours: Tuesdays from 13:00-14:00 in the Learning cafe (School Social Hub (Room MB-B11)). 



  • Syllabus

    The module will cover the following general topics:

    1. Vector algebra and calculus
      • Tangent vectors, calculus of vector-valued functions
    2. The geometry of curves
      • Parametrisations, tangent lines, orientation, curve integrals
    3. The geometry of surfaces
      • Parametrisations, tangent planes, orientation, surface integrals
    4. Applications
      • Constrained optimisation and Lagrange multipliers; Green's, Stokes', and divergence theorems

  • Learning Outcomes

    At the end of this module, students should be able to:

    • Precisely define and describe basic geometric concepts, such as curves and surfaces.
    • Explain how geometric properties of curves and surfaces and be captured and quantified, and apply apply this knowledge in order to perform computations.
    • Demonstrate how ideas from differential and integral calculus can be extended to geometric settings.
    • Demonstrate how calculus and linear algebra can be connected to and applied to geometric concepts and questions.
    • Explain the main ideas behind proofs of basic results in curve and surface geometry.
    • Explain some ways that geometric ideas can be applied to questions in other areas of mathematics or outside of mathematics.

    • File icon
      Index of Core Skills File
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      This spreadsheet contains a list of all the "core skills" for the module that you should know. Also, next to each listed skill are examples and problems from the module where this skill was primarily used.

      While this is not an exhaustive list of everything that could be examined, it is quite extensive, and you will pass the module very comfortably if you can carry out and explain all the things on this list.


  • Module Description

    This module develops the language and tools for studying, describing and quantifying the geometry of curved objects. Particular emphasis is placed on connecting geometric questions with ideas from Calculus and Linear Algebra, as well as on extending Calculus to curved settings. The module concludes by studying some landmark results in vector Calculus e.g. Lagrange multipliers, Green's theorem and Stokes' theorem.

  • Week 1

    Content covered: Sections 2.1, 2.2, 2.3, 2.4 of Lecture Notes.


  • Week 2

    Content covered: Sections 2.5, 2.6, 2.7, 2.8 of Lecture Notes.


  • Week 3

    Content covered: Sections 2.9, 2.10, 3.1, 3.2 of Lecture Notes.


  • Week 4

    Content covered: Sections 3.3, 3.4, 3.5 of Lecture Notes.


  • Week 5

    Content covered: Sections 3.6, 3.7, 3.8, 3.9 of Lecture Notes.


  • Week 6

    Content covered: Sections 3.10, 4.1, 4.2 of Lecture Notes


  • Week 7

    Content covered: Sections 4.3, 4.4 of Lecture Notes.


  • Week 8

    Content covered: Sections 4.4, 4.5, 4.6, 4.7 of Lecture Notes.


  • Week 9

    Content covered: Sections 4.8, 4.9, 4.10, (5.1) of Lecture Notes.


  • Week 10

    Content covered: Sections (finish 4.10), 5.1, 5.2, 5.3, 5.4 of Lecture Notes.


  • Week 11

    Content covered: Sections 5.5 and 5.6 of Lecture Notes.


  • Week 12

    Content covered: Sections 5.7, 5.8, 5.9 of Lecture Notes.


  • Assessment information

    • Assessment Pattern -  

      The assessments for this module will consist of:

      • Two courseworks, each counting for 10% of your module mark. These will be due in Weeks 6 and 12.
      • The final exam in May, which will count for 80% of your module mark.

      Format and dates for the in-term assessments - 

      In weeks 6 and 12 you will be asked to hand in coursework

      Format of final assessment - 

      Your final examination will be on campus with location TBD.  It will be 3 hours in duration with SpLD accommodations handled separately.  No calculators will be allowed. 

      Link to past papers -

      Link to past paper repository 

      Description of Feedback - 

      You can discuss with me about how you're doing

    • File icon
      Sample question and good vs bad answers File
      168.6 KB

  • Assessment

    • The assessments for this module will consist of:

      • Two courseworks, each counting for 10% of your module mark. These will be due in Weeks 6 and 12.
      • The final exam in May, which will count for 80% of your module mark.


  • Coursework

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  • Exam papers

  • Q-Review

  • Early feedback questionnaire

  • Online Reading List

  • Graphical Tools