Multivariable Analysis
Faculteit  Science and Engineering 
Jaar  2020/21 
Vakcode  WBMA02205 
Vaknaam  Multivariable Analysis 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  https://rooster.rug.nl/#/en/current/course/WBMA19011 
Uitgebreide vaknaam  Multivariable Analysis  
Leerdoelen  At the end of the course, the student is able to: 1. Explain how the assumption of differentiability allows us to extend notions of (multi)linear algebra locally, in an infinitesimal setting. In particular simple linear algebra proofs and constructions should be understood at a sufficiently deep level so as to be able to investigate, modify and apply them in an analysis context. 2. Assess the existence and uniqueness of a system of (differential) equations. This includes being able to apply the general solution techniques to find solutions in simple practical situations. 3. Define differential forms and understand their use in integration. Moreover one should be able to apply Stokes theorem and calculate exterior derivatives and integrals. 4. Reproduce the key definitions and theorems in this field: inverse and implicit function theorem, Picard theorem, Stokes theorem, Change of variables theorem and Poincare lemma. The main ideas of the proofs of these theorems should be clear. 

Omschrijving  The goal of this course is to explore the notions of differentiation and integration in arbitrarily many variables. The material is focused on answering two basic questions: 1) How to solve an equation? How many solutions can one expect? 2) Is there a higher dimensional analogue the fundamental theorem of calculus? Can one find a primitive? The equations we will address are systems of nonlinear equations in finitely many variables and also ordinary differential equations. The approach will be mostly theoretical, sketching a framework in which one can predict how many solutions there will be without necessarily solving the equation. The key assumption is that everything we do can locally be approximated by linear functions. In other words, everything will be differentiable. One of the main results is that the linearisation of the equation predicts the number of solutions and approximates them well locally. This is known as the implicit function theorem. For ordinary differential equations we will prove a similar result on the existence and uniqueness of solutions. To introduce the second question, recall what the fundamental theorem of calculus says that the integral of the derivative is the same as the 'integral' of the function on the boundary. What if our function depends on two or more variables? In two and three dimensions, vector calculus gives some partial answers involving div, grad, curl and the theorems of Gauss, Green and Stokes. How can one make sense of these and are there any more such theorems perhaps in higher dimensions? The key to understanding this question is to pass from functions to differential forms. In the example above this means passing from f(x) to the differential form f(x)dx. Taking the dx part of our integrands seriously clarifies all formulas and shows the way to a general fundamental theorem of calculus that works in any dimension. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(There will be 7 homework sets and the final grade is computed as max(T,(H+3T)/4) where H is the average of the best six homework sets and T is the final exam.) 

Vaksoort  bachelor  
Coördinator  dr. R.I. van der Veen  
Docent(en)  dr. R.I. van der Veen  
Verplichte literatuur 


Entreevoorwaarden  This course assumes prior knowledge of: Linear Algebra 1,2, Calculus 1,2 and Metric and Topological Spaces  
Opmerkingen  This course was registered last year with course code WBMA19011  
Opgenomen in 
