Course MA2322

This is a one-semester course on Calculus on manifolds, to be given in Hilary term 2015.

It follows on from course MA2321 given in Michaelmas term 2014. on functions of several real variables. The two courses, MA2321 and MA2322 correspond essentially to the one year long course 224, which was given in 2006-2007, 2007-2008 and 2008-2009.

A (smooth) manifold is a space which local coordinates can be introduced which map onto open sets in R^n, and such that changes of coordinates are given by differentiable functions. In this course we first develop the essential facts about tensors (scalar valued multilinear functions) in a linear algebra context, before applying these ideas to differentiation and integration on a manifold, leading up to the theorems of Stokes and Poincare.

Most of the topics in courses MA2321 and MA2322 are covered in the notes of two earlier one-year courses, 211 and 221, which are available as pdf files at:

PDF file for Course 211 (Linear algebra and Differential Forms).

PDF file for Course 221 (Real and Complex Analysis) .

See in particular sections 3, 4, 5, 6, 7.3, 9, 11 and 12 of course 211

and sections 4, 6 and 7 of course 221.

The objective of this course is to familiarise you with the basic facts on integration and differentiation on smooth manifols, and the associated geometrical ideas and concepts of multilinear algebra.

The emphasis is on the theory, and the applications are intended to strengthen the grasp of the theory.

The course is not based on any specific books. The webnotes mentioned above may form useful background reading.

The following, however, may be of interest as further background reading:

Advanced Calculus: a differential forms approach by Harold M. Edwards

Calculus on Manifolds by Michael Spivak

PDF file for Exercises 1 .

PDF file for Exercises 2 .

PDF file for Exercises 3 .

PDF file for Exercises 4 .

PDF file for Exercises 5 .

My office telephone is +353-1-896 1944 and the departmental fax is at +353-1-896 2282