Analysis
Faculteit  Science and Engineering 
Jaar  2017/18 
Vakcode  WPMA14004 
Vaknaam  Analysis 
Niveau(s)  propedeuse 
Voertaal  Engels 
Periode  semester II b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Analysis  
Leerdoelen  1. The student is able to reproduce basic concepts and constructs from the theory of real numbers such as the axiom of completeness, density of the set of rational numbers, and Cantor’s diagonalization method. The student is able to rigorously formulate logical statements with quantifiers such as ‘for all’ and ‘there, and to formulate their negations. 2. The student is able to review and reproduce the basic concepts and theorems of the theory of sequences and series, to give formal and rigorous mathematical proofs for various consequences of this theory, and to construct counterexamples for nonvalid statements. 3. The student is able to reproduce basic concepts and theorems from the topological theory of the set of real numbers, using notions like ‘open’, ‘closed’, ‘dense’, ‘compact’, and ‘connected’. The student is able to apply this theory to examples, and formally verify such properties in examples. The student is able to rigorously prove simple corollaries of this theory, or to produce counterexamples for nonvalid statements. 4. The student is able to reproduce the basic theory of continuity and differentiability of functions of one variable, including the extreme value, intermediate value and mean value theorems. The student is able to give formal ‘epsilondelta’ proofs of consequences of this theory, and to apply this theory to examples. 5. The student is able to reproduce the formal reasoning and theorems of the theory of sequences and series of functions, and to be able to apply this theory to examples. 6. The student is able to reproduce the basic theory of Riemann integrals, and to apply this to examples. 

Omschrijving  This course provides a fundamental mathematical underpinning of many of the concepts and techniques as treated in the courses Calculus 1 and 2. Contrary to the courses Calculus 1 and 2, which focus on the computational skills and the applicability to natural and engineering sciences and beyond, the course on Analysis treats the underlying mathematical fundaments and their full proofs. In particular, the course provides an introduction to the theory of the real numbers, with concepts like ‘infimum’, ‘supremum’, ‘complete‘, ‘(un)countable‘, ‘open’, ‘closed’, ‘dense’,‘compact’, and ‘connected’. Furthermore, the course recalls and further formalizes the concepts of ‘limit’ of sequences and series, including the Cauchy criterion, and ‘continuity’ and ‘differentiability’ of functions of a single variable. One of the main aims of the course is to provide a sound introduction and training to conducting mathematical proofs, including the socalled ‘epsilondelta’ reasoning, or otherwise being able to construct counterexamples. Classical theorems such as the ‘intermediatevalue theorem’ and the ‘meanvalue theorem’ are formulated and proved in full detail. Uniform convergence of sequences of functions, as well as continuity and differentiability of the limit function, is treated in detail. This is applied to power and Taylor series, in much more mathematical rigor than in the calculus courses. The course finally provides the basic theory of Riemann integrals, culminating in the ‘fundamental theorem of calculus’. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(Assessment takes place through four homework assignments and a final written exam. The final grade is obtained by taking the following weighted average: the average of the homework assignments counts for 30%, and the grade of the final exam counts for 70%. The same weighted average applies if a reexam is taken. However, if the grade of the (re)exam is larger than the weighted average, then the final grade is taken equal to the grade of the (re)exam.) 

Vaksoort  propedeuse  
Coördinator  dr. A.E. Sterk  
Docent(en)  dr. A.E. Sterk  
Verplichte literatuur 


Entreevoorwaarden  
Opmerkingen  Note: in June 2015 Springer has published a second edition of the book. We will use this second edition.  
Opgenomen in 
