Analysis
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WBMA01205 
Vaknaam  Analysis 
Niveau(s)  propedeuse 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Analysis  
Leerdoelen  At the end of the course, the student is able to: 1. 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; rigorously formulate logical statements with quantifiers such as ‘for all’ and ‘there, and formulate their negations. 2. reproduce the basic concepts and theorems of the theory of sequences and series, give formal and rigorous mathematical proofs for various consequences of this theory, and construct counterexamples for nonvalid statements. 3. reproduce basic concepts and theorems from the topological theory of the set of real numbers, using notions like ‘open’, ‘closed’, ‘dense’, ‘compact’, and ‘connected’; apply this theory to examples, and formally verify such properties in examples; rigorously prove simple corollaries of this theory, or produce counterexamples for nonvalid statement 4. reproduce the basic theory of continuity and differentiability of functions of one variable, including the extreme value, intermediate value and mean value theorems; give formal ‘epsilondelta’ proofs of consequences of this theory, and to apply this theory to examples. 5. reproduce the formal reasoning and theorems of the theory of sequences and series of functions, and apply this theory to examples. 6. reproduce the basic theory of Riemann integrals, and apply this to examples. 

Omschrijving  This course provides the fundamental mathematical underpinning of many 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‘ness, ‘and (un)countable, ‘open’, ‘closed’, ‘dense’,‘ compact’, and ‘connected’ sets. Furthermore, the course recalls and further formalizes the concepts of ‘limit’s 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 writing mathematical proofs, including the socalled ‘epsilondelta’ reasoning, or otherwise the ability 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, which culminates in the ‘fundamental theorem of calculus’. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(Final Grade = max(WE, 0.3 HW + 0.7 WE) only if WE >=4.5, otherwise Final Grade = WE, where HW average homework assignments grade and WE grade final written exam. The same formula applies to the resit exam.) 

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


Entreevoorwaarden  The course unit assumes prior knowledge acquired from Calculus 1.  
Opmerkingen  
Opgenomen in 
