Analysis on Manifolds

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, Metric Spaces and Dynamical Systems 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 (coordinate-free, global analysis)
Omschrijving	This course unit will introduce curved spaces and generalize the concepts of calculus to such spaces. The center of attention with be 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. Introducing such constructions, students will learn how lots of calculus concepts can be reintroduced in a "coordinate-free" manner. 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. • Applications to Topology and Mechanics. Applications are adapted each year to the interests of the class and include an introduction to the De Rham cohomology and the Brouwer degree, fixed-point theorems and the hairy ball theorem, integrability and Frobenius theorem, distributions, connections on vector bundles. Ultimately, throughout the course, we develop a mathematical language that constitutes the basis of large parts of what happens in more advanced topics in mathematics and mathematical physics.
Uren per week
Onderwijsvorm	Hoorcollege (LC), Opdracht (ASM), 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	dr. M. Seri
Docent(en)	dr. M. Seri
Verplichte literatuur	Titel Auteur ISBN Prijs An Introduction to Manifolds, Springer, 2010 Loring W. Tu 978-1-4419-7400-6 Introduction to Smooth Manifolds. Springer Springer GTM 218. Institutional access available via SpringerLink DOI: https://doi.org/10.1007/978-1-4419-9982-5 ebook ISBN: 978-1-4419-9982-5 2012 J.M. Lee 978-1-4419-9981-8
Entreevoorwaarden	The course build on knowledge of Calculus 1 & 2, Linear Algebra 1 & 2, Multivariable Analysis, Topology and Metric Spaces.
Opmerkingen
Opgenomen in	Opleiding Jaar Periode Type BSc Applied Mathematics 3 semester I b keuze BSc Courses for Exchange Students: AI - Computing Science - Mathematics - semester I b BSc Mathematics and Physics (double degree) 3 semester I b keuzegroep BSc Mathematics: General Mathematics  (Electives and Minor General Mathematics) 3 semester I b keuze BSc Mathematics: General Mathematics  ( Major track General Mathematics) 3 semester I b keuzegroep BSc Mathematics: Probability and Statistics 3 semester I b keuze