Multivariable Analysis

Uitgebreide vaknaam	Multivariable Analysis
Leerdoelen	At the end of the course, the student is able to: 1. define the notions of differentiability and integrability in the multi-dimensional setting, compute the differentiability class of specific functions, use Fubini's theorem to compute specific multiple integrals; 2. translate the concepts of linear algebra, such as vectors, convectors and alternating multilinear maps, to the differentiable case. Explain the notion of differential forms and define standard operations on these forms (exterior product, pullback, etc.); 3. state Banach's fixed point theorem and use it to prove Picard-Lindelöf's and the implicit function theorems; 4. assess the existence and uniqueness of a system of implicit equations as well as ordinary differential equations. In the parametric case, this includes being able to verify a smooth dependence of the solution on the parameters; 5. explain the role of differential forms in integration. Reproduce Stokes's theorem and deduce theorems of Green and Gauss from this theorem. This includes being able to apply these theorems in specific examples.
Omschrijving	The goal of this course is to explore the notions of differentiation and integration in arbitrarily many variables. The material is focused on extending the corresponding theory of one variable to the multi-dimensional case and understanding the new phenomena that arise in this more general setting. On the differentiation side, we will in particular discuss the notions of total derivative, partial and directional derivatives, the Jacobian matrix, and on the integration side --- multiple integration and differential forms. We will also touch upon infinitedimensional spaces and in particular the Banach fixed point theorem, which we will use to derive the implicit function theorem and the Picard-Lindelöf existence and uniqueness theorem for ordinary differential equations. Some of the main results that we will prove in this course are the following: - Schwarz's and Fubini's theorems (on changing the order of differentiation/integration); - Banach's fixed point theorem and its applications (the inverse and the implicit function theorems, Picard-Lindelöf's theorem); - Stokes's theorem and its applications (theorems of Green and Gauss, Brouwer's fixed point theorem).
Uren per week
Onderwijsvorm	Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
Toetsvorm	Opdracht (AST), Schriftelijk tentamen (WE) (The final grade is computed as max(E,0.7E+0.3H), where E is the exam's grade and H is the average of the 3 homework sets. If E is 4 or less, the final grade is E.)
Vaksoort	bachelor
Coördinator	dr. N. Martynchuk
Docent(en)	dr. N. Martynchuk
Verplichte literatuur	Titel Auteur ISBN Prijs Recommended: Mathematical Analysis 1 Vladimir A. Zorich 978-3-540-87451-5 Recommended: Calculus on Manifolds, a modern appreach to classical theorems of advanced calculus (1965) Michael Spivak 0-8053-9021-9 Readers: Last year lecture notes available from the website http://www.rolandvdv.nl/MA Roland van der Veen Mandatory: Real Mathematical Analysis 2 (2015) Charles C. Pugh 978-3-319-17771-7
Entreevoorwaarden	This course assumes prior knowledge of: Linear Algebra 1,2 and Calculus 1,2
Opmerkingen	This course was registered last year with course code WBMA19011
Opgenomen in	Opleiding Jaar Periode Type BSc Mathematics and Physics (double degree) 2 semester I b verplicht BSc Mathematics: General Mathematics  ( Major track General Mathematics) 2 semester I b verplicht BSc Mathematics: Probability and Statistics 3 semester I b keuze BSc Technische Wiskunde 2 semester I b verplicht