Fields, Forms and Flows 3

Unit aims

To introduce the main tools of the theory of vector fields, differential forms and flows.

Unit description

A differentiable manifold is a space which looks locally like Euclidean space but which globally may not. Familiar examples include spheres, tori, regular level sets of functions f(x) on Rn , the group of invertible n x n matrices.

In the unit we develop the theory of vector fields, flows and differential forms mainly for Rn but with a view towards manifolds, in particular surfaces in R3.

The theory of differentiable manifolds extends ideas of calculus and analysis on Rn to these non-Euclidean spaces. An extensive subject in its own right, the theory is also basic to many areas of mathematics (eg, differential geometry, Lie groups, differential topology, algebraic geometry) and theoretical physics and applied mathematics (eg, general relativity, string theory, dynamical systems). It is one of the cornerstones of modern mathematical science.

Important elements in the theory are i) vector fields and flows, which provide a geometrical framework for systems of ordinary equations and generalise notions of linear algebra, and ii) differential forms and the exterior derivative. Differential forms generalise the line, area and volume elements of vector calculus, while the exterior derivative generalises the operations of grad, curl and divergence. The calculus of differential forms generalises and unifies a number of basic results (eg, multidimensional generalisations of the fundamental theorem of calculus: Green's theorem, Stokes' theorem, Gauss's theorem) whilst at the same time bringing to light their geometrical aspect.

Relation to other units

The material on Stokes' theorem is relevant to simplicial homology, which is treated in the Level 4 unit Algebraic Topology from a different point of view.  The unit complements material in Introduction to Geometry, Topics in Modern Geometry 3 and Lie groups, Lie algebras and their representations.

Learning objectives

At the end of the unit students should:

  • Know and understand the definition of vector fields and flows; be able to calculate flows for simple examples.
  • Know and understand the definition of the Jacobi bracket, be able to derive its properties and compute it in examples.
  • Know and understand Frobenius integrability theorem and its proof, and be able to apply it to systems for first order PDE's
  • Have facility with the algebra and calculus of differential forms, including the wedge product and exterior derivative
  • Know and understand the Poincaré lemma and its proof, and be able to apply it
  • Know and understand Stokes' theorem for singular cells and its proof; be able to apply it; be familiar with its extension to manifolds.
  • Be able to apply the material in the unit to unseen situations

Transferable skills

Mathematical skills: Knowledge of differentiable manifolds; facility with differential forms, tensor calculus; geometrical reasoning

General skills: Problem solving and logical analysis; Assimilation and use of complex and novel ideas


  • Vector fields, flows and diffeomorphisms [~5 weeks]: Maps and diffeomorphisms on Rn. ODE's, vector fields, and flows.  Jacobi bracket. Frobenius integrability theorem.
  • Algebraic k-forms and differential forms [~4 weeks]: Algebraic k-forms. Wedge product. Differential forms, Exterior derivative. Poincaré Lemma.
  • Stokes' Theorem [~1 week]: Integration of forms. Cells and boundaries. Stokes' Theorem.

Reading and References

Typeset lecture notes will be provided.  There is no single required or recommended text.  The following are standard texts that cover some or all of the material.  More information on these references is provided in the lecture notes.

  1. JH and BB Hubbard, "Vector calculus, linear algebra and differential forms: A unified approach", 2 ed, Prentice Hall
  2. B Schutz, Geometrical methods in mathematical physics, Cambridge University Press
  3. W Darling, Differential forms and Connections, Cambridge University Press
  4. M Spivak, A comprehensive introduction to differential geometry, vol 1, Publish or Perish, Berkeley.
  5. V Arnold, Mathematical methods of classical mechanics, Springer-Verlag.

Unit code: MATH30018
Level of study: H/6
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Dr Martin Sieber


MATH20101 Ordinary Differential Equations 2 plus either MATH20901 Multivariable Calculus or MATH20006 Metric Spaces



Methods of teaching

Lectures, lecture notes, problem sheets, solutions.

Methods of Assessment 

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% from a 2 hour 30 minute exam

NOTE: Calculators are NOT allowed in the examination. 

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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