Analysis on Manifolds
Faculteit  Science and Engineering 
Jaar  2019/20 
Vakcode  WIANVAR07 
Vaknaam  Analysis on Manifolds 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Analysis on Manifolds  
Leerdoelen  1. Be able to work with differentiable manifolds in an abstract setting, and with vector fields and differential forms on manifolds . 2. Have knowledge of and be able to apply the Poincaré Lemma and the Stokes formula 3. Be able to make connections with earlier content from Multivariate Calculus, Linear Algebra, and Mathematical Physics 4. Know and be able to prove abstract theorems (for example the Poincaré Lemma, Stokes’ theorem) 5. Recognize and be able to apply these abstract theorems in concrete situations (for example be able to deduce Green’s and Cauchy’s Theorems from Stokes’ Theorem, apply Poincaré’s Theorem when solving exact differential equations) 6. Be able to apply proof techniques to demonstrate basic results from the lectures (coordinatefree, global analysis) 

Omschrijving  This course unit will generalize concepts such as curves and surfaces to the general concept of differential manifold. Each point in such a manifold has a tangent space with the structure of a vector space and which hangs off the point of application in a ‘smooth’ manner. The dual space of this is the cotangent space. The following will be addressed: • Vector fields and differential forms and their transformation rules when transferring to different local parameterization of the manifold. • Integration of differential forms on orientable manifolds and the Stokes’ theorem. • The concepts of exact and closed applied to differential forms, whereby several versions of the Poincaré Lemma occur. This theory has applications in both mathematical physics as well as in geometry (topology). The latter will include an introduction to the De Rham cohomology and the Brouwer degree. Applications include the fixedpoint theorem and the hairy ball theorem: both wellknown results produced by Brouwer. Thus, a mathematical language develops that constitutes the basis of large parts of what happens in more advanced mathematics and mathematical physics. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(Assessment takes place through 3 homework assignments and a written exam. The final grade is obtained by taking the following weighted average: the average of the 3 homework assignments counts for 30% and the grade of the final exam counts for 70%. In case the grade of the final exam is larger than this weighted average the final grade is just the exam grade.) 

Vaksoort  bachelor  
Coördinator  prof. dr. G. Vegter  
Docent(en)  prof. dr. G. Vegter  
Verplichte literatuur 


Entreevoorwaarden  the course build on knowledge of Calculus 1 & 2, Linear Algebra 1 & 2, Metric Spaces  
Opmerkingen  
Opgenomen in 
