Analysis 1

Introduction: numbers, sequences, cluster and limit points, recursive sequences.

Functions: functions of one and several variables, level curves, level surfaces, basic properties, continuity, limit values, properties of continuous functions.

Differential calculus of one variable: definition and geometric property, differentiation rules, derivatives of elementary functions, properties of differentiable functions, applications in analysing functions (graphs, limits, extremal points), Taylor's formula.

Differential calculus of several variables: partial derivatives, gradient and directional derivatives, differentiability, Taylor's formula, local and constrained extrema, implicit function theorem.

Primitive function and indefinite integral.

I. Vidav: Višja Matematika I, DMFA-založništvo, Ljubljana, 1994.
J. Stuart: Calculus, 5th edition, Brooks/Cole, Pacific Grove, 2003.
K. A. Ross: Elementary analysis: The theory of calculus, New York-Heidelberg, 2003.
S. Lang: Calculus of Several Variables, 3rd edition, Springer, New York, 1996.
G. N. Berman: A problem book in mathematical analysis, Moskva, Mir Publ. , 1980.
G. Tomšič, B. Orel, N. Mramor Kosta: Matematika I, Založba FE in FRI, Ljubljana, 2004.
G. Tomšič, B. Orel, N. Mramor Kosta: Matematika II, Založba FE in FRI, Ljubljana, 2005.
P. Mizori-Oblak: Matematika za študente tehnike in naravoslovja I, Fakulteta za strojništvo, Ljubljana, 2001.
P. Mizori-Oblak: Matematika za študente tehnike in naravoslovja I, Fakulteta za strojništvo, Ljubljana, 2003.

Students learn the basic concepts and techniques of mathematical analysis, such as function, limit, continuity and derivative. Analysis 1 is a fundamental course for other mathematical and special courses in study of mathematics, natural science, technical science and other fields of science.

Knowledge and understanding: Understanding of real functions of one and several variables and of differential calculus.

Application: Analysis 1 is a fundamental course in natural science, technical science, social science and other fields of science.

Reflection: Understanding of the theory from the examples and applications.

Transferable skills: The ability to formulate a problem in mathematical language, the ability to select an appropriate method. The ability to accurately solve the problem and analyse the obtained results.

Lectures, exercises, homeworks, consultations

Type (examination, oral, coursework, project):

2 midterm exams instead of problem-based exam, problem-based exam

theoretical knowledge exam

grading: 5 fail, 6-10 pass (according to the Statute of UL)

Miran Černe:
– ČERNE, Miran, FLORES, Manuel. Boundary value problems for holomorphic functions on the upper half-plane. The Asian journal of mathematics, ISSN 1093-6106, 2007, vol. 11, no. 4, str. 609-620 [COBISS-SI-ID 14762073]
– 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, 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. 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]
Oliver Dragičević:
– DRAGIČEVIĆ, Oliver. Weighted estimates for powers of the Ahlfors-Beurling operator. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 6, str. 2113-2120 [COBISS-SI-ID 15876697]
– DRAGIČEVIĆ, Oliver. Some remarks on the L [sup] p estimates for powers of the Ahlfors-Beurling operator. Archiv der Mathematik, ISSN 0003-889X, 2011, vol. 96, no. 5, str. 463-471 [COBISS-SI-ID 16090713]
– DRAGIČEVIĆ, Oliver, VOLBERG, Alexander. Linear dimension-free estimates in the embedding theorem for Schrödinger operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 85, p. 1, str. 191-222 [COBISS-SI-ID 16214873]
Sašo Strle:
– STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511. [COBISS-SI-ID 13135193]
– GRIGSBY, J. Elisenda, RUBERMAN, Daniel, STRLE, Sašo. Knot concordance and Heegaard Floer homology invariants in branched covers. Geometry & topology, ISSN 1364-0380, 2008, vol. 12, iss. 4, str. 2249-2275 [COBISS-SI-ID 14892121]
– LEVINE, Adam Simon, RUBERMAN, Daniel, STRLE, Sašo, GESSEL, Ira M. Nonorientable surfaces in homology cobordisms. Geometry & topology, ISSN 1465-3060, 2015, vol. 19, no. 1, str. 439-494. [COBISS-SI-ID 17557337]
– DRINOVEC-DRNOVŠEK, Barbara, STRLE, Sašo. Naloge iz analize 1 : z odgovori, nasveti in rešitvami, (Izbrana poglavja iz matematike in računalništva, 46). 2. izd. Ljubljana: DMFA - založništvo, 2016. 285 str., ilustr. ISBN 978-961-212-251-5 [COBISS-SI-ID 287122944]