Unit information: Fields, Forms and Flows in 2020/21

Unit name	Fields, Forms and Flows
Unit code	MATH30018
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Sieber
Open unit status	Not open
Pre-requisites	Year 2 Theoretical Physics OR MATH20100 Ordinary Differential Equations 2 plus either MATH20901 Multivariable Calculus or MATH20006 Metric Spaces.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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 Algebraic Topology from a different point of view. The unit complements material in Introduction to Geometry, Topics in Modern Geometry 3 and Algebraic Geometry.

Intended Learning Outcomes

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

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Recommended

• V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1989
• R.W.R Darling, Differential Forms and Connections, Cambridge University Press, 1994
• John H. Hubbard and Barbara B. Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach, 2 ed, Prentice Hall, 2009
• Bernard F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, 1980
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, vol 1, Publish or Perish, Berkeley, 2005