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 |
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.
Learning Objectives
At the end of the unit students should:
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
The unit will be taught through a combination of
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.
Recommended