Introduction to manifolds
Collegejaar:  20112012 

Studiegidsnummer:  IMF 
Docent(en): 

Voertaal:  Engels 
Blackboard:  Onbekend 
EC:  6 
Niveau:  400 
Periode:  Semester 1 
 Geen Keuzevak
 Geen Contractonderwijs
 Wel Exchange
 Wel Study Abroad
 Geen Avondonderwijs
 Geen AlaCarte en Aanschuifonderwijs
 Geen Honours Class
This is a course on DIFFERENTIABLE MANIFOLDS, the abstract generalization of smooth curves and surfaces in euclidean space. Such a manifold has a topology and a certain dimension n, and locally it is homeomorphic with a piece of ndimensional euclidean space, such that these pieces (in general in a nontrivial way) “are differentiably glued together”. Examples are the ndimensional sphere and projective space, and ndimensional smooth quadratic hypersurfaces in (n+1)dimensional euclidean space; the latter are the ndimensional analogues of the smooth quadratic curves and surfaces treated in linear algebra 2.
In particular, it is possible to define differentiable functions on and differentiable maps between differentiable manifolds.
A crucial ingredient in this theory is the TANGENT SPACE of a manifold at a point, the generalization of the tangent line resp. tangent plane of a curve resp. surface in euclidean space; this concept enables one to define a kind of Jacobian of a smooth map between manifolds, and to generalize the Inverse and Implicit Function Theorem.
Using the dual of the tangent space we further introduce DIFFERENTIAL FORMS which are used for integration on manifolds. Here the aim is to prove the THEOREM OF STOKES, the general version of the classical integral theorems of Gauss and Greene.
Prerequisites
Differentiation and integration of functions of several real variables; Topology
Literature
Reader, downloadable via the link below
Recommended book
K. Jänich; Vector analysis, ISBN 0387986499, Springer Verlag 2001
Examination
1/3 homework, 2/3 oral exam
Links
homepage of the course
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Mathematics  Master  1  
Wiskunde  Bachelor  1 