Analysis 1

2022/2023
Programme:
Mathematics Education
Year:
1 year
Semester:
first and second
Kind:
mandatory
ECTS:
18
Language:
slovenian
Hours per week – 1. semester:
Lectures
4
Seminar
0
Tutorial
4
Lab
0
Hours per week – 2. semester:
Lectures
4
Seminar
0
Tutorial
4
Lab
0
Content (Syllabus outline)

Number systems. Field axioms. Order. Supremum. Dedekind's axiom. Euclidean spaces. Maps and functions. Cardinality. Cardinal numbers.
Numerical sequences. Cluster and limit points. Cauchy criterion. Algebraic operations on convergent sequences. Monotonic sequences. Elementary theory on numerical series. Convergence. Cauchy convergence test. Algebraic operations. Convergence tests. Power series. Absolute and conditional convergence. Leibniz criterion. Cauchy product of series.
Functions of one variable. Continuity. Limit values. Properties of continuous functions. Overview of continuity of elementary functions.
Derivative of a function. Geometric and physical meaning. Differentiation rules. Rolle's theorem, Langrage's mean value theorem. Applications in analysing functions. Higher derivatives. L'Hôpital's rule.
Indefinite and definite integral of function of one variable. Theorems on the existence of an integral. Properties of integral. Relation between definite and indefinite integral. Mean value theorem. Improper integrals. Applications. Integration methods.
Convergence of functional sequences and series. Term-by-term differentiation and integration. Taylor series, remainder and applications. Series of particular elementary functions.
Topology of metric spaces. Cluster points of sets and sequences. Limit of a sequence. Open and closed sets. Cauchy sequences and complete metric spaces. Compactness. Heine-Borel theorem. Continuous maps. Banach fixed-point theorem.

Readings

I. Vidav: Višja Matematika I, DMFA-založništvo, Ljubljana, 1994.
N. Prijatelj: Uvod v matematično analizo I, DMFA-založništvo, Ljubljana, 1980.
S. Lang: Undergraduate Analysis, 2nd edition, Springer, New York, 1997.
W. Rudin: Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, New York-Auckland-Düsseldorf, 1976.
R. S. Strichartz: The Way of Analysis, Jones & Bartlett, Boston, 2000.
K. A. Ross: Elementary Analysis : The Theory of Calculus, Springer, New York-Heidelberg, 2003.

Objectives and competences

Students learn the basic concepts of mathematical analysis such as limit, continuity, derivative and integral of real functions of one variable, numerical and function series, and metric spaces. Analysis 1 is one of the fundamental courses in mathematics.

Intended learning outcomes

Knowledge and understanding: Understanding of differential and integral calculus of real functions of one variable and related topics. The application of method in geometry and natural science.
Application: Analysis 1 is one of the fundamental courses in mathematics, it can be used in natural science and other field of science
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to design the problem, select an appropriate method, solve the problem, and analyse the results on test cases. The ability to formulate a problem in mathematical language. Skills in using the literature.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Type (examination, oral, coursework, project):
4 midterm exams instead of written exam, written exam
oral exam
grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Miran Černe:
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. Some remarks on Hartogs' extension lemma. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2010, vol. 138, no. 10, str. 3603-3609. [COBISS-SI-ID 15696473]
ČERNE, Miran. Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces. American journal of mathematics, ISSN 0002-9327, 2004, vol. 126, no. 1, str. 65-87. [COBISS-SI-ID 12895577]
ČERNE, Miran. Matematika 2, (Matematični rokopisi, 24). Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije: DMFA - založništvo, 1999. 127 str., ilustr. ISBN 961-212-096-X. [COBISS-SI-ID 103971072]
Barbara Drinovec Drnovšek:
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Strongly pseudoconvex domains as subvarieties of complex manifolds. American journal of mathematics, ISSN 0002-9327, 2010, vol. 132, no. 2, str. 331-360. [COBISS-SI-ID 15549529]
DRINOVEC-DRNOVŠEK, Barbara. Discs in Stein manifolds containing given discrete sets. Mathematische Zeitschrift, ISSN 0025-5874, 2002, vol. 239, no. 4, str. 683-702. [COBISS-SI-ID 11567449]
DRINOVEC-DRNOVŠEK, Barbara. Proper holomorphic discs avoiding closed convex sets. Mathematische Zeitschrift, ISSN 0025-5874, 2002, vol. 241, no. 3, str. 593-596. [COBISS-SI-ID 12076377]
DRINOVEC-DRNOVŠEK, Barbara, STRLE, Sašo. Naloge iz analize 1 : z odgovori, nasveti in rešitvami, (Izbrana poglavja iz matematike in računalništva, 46). 1. natis. Ljubljana: DMFA - založništvo, 2010. 285 str., ilustr. ISBN 978-961-212-219-5. [COBISS-SI-ID 250561280]
Franc Forstnerič:
FORSTNERIČ, Franc. Holomorphic families of long c [sup] 2's. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2012, vol. 140, no. 7, str. 2383-2389. [COBISS-SI-ID 16435289]
FORSTNERIČ, Franc. Stein manifolds and holomorphic mappings : the homotopy principle in complex analysis, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, vol. 56). Heidelberg [etc.]: Springer, cop. 2011. X, 489 str., ilustr. ISBN 978-3-642-22249-8. ISBN 978-3-642-22250-4. [COBISS-SI-ID 16008025]
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]