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Analysis on Manifolds
| Faculteit | Science and Engineering |
| Jaar | 2022/23 |
| Vakcode | WBMA013-05 |
| 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 | At the end of the course, the student is able to: 1. 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. make connections with earlier content from Multivariate Calculus, Linear Algebra, Metric Spaces and Dynamical Systems 4. know and prove abstract theorems (for example the Poincaré Lemma, Stokes' theorem) 5. recognize and 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. apply proof techniques to demonstrate basic results from the lectures (coordinate-free, global analysis) |
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| Omschrijving | This course unit will introduce curved spaces and generalize the concepts of calculus to such spaces. The centre 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. |
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| 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 F is computed as follows: F = if E>4 then max(30% * H + 70% * E, E) else E. Here, E is the grade of the final exam and H is the following average of the homework assignments: if the grades of **all** homework assignments are >4, then H="average of best 3 assignments" else H="average of all assignments".) |
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| Vaksoort | bachelor | ||||||||||||||||||||||||||||
| Coördinator | Dr. M. Seri | ||||||||||||||||||||||||||||
| Docent(en) | Dr. M. Seri | ||||||||||||||||||||||||||||
| Verplichte literatuur |
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| Entreevoorwaarden | Knowledge assumed: Calculus 1 & 2, Linear Algebra 1 & 2, Multivariable Analysis, Topology and Metric Spaces. | ||||||||||||||||||||||||||||
| Opmerkingen | The notes will be made available on Nestor and are always available for free at https://github.com/mseri/AoM/releases | ||||||||||||||||||||||||||||
| Opgenomen in |
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