Mathematics 2

2022/2023
Programme:
Practical Mathematics
Year:
2 year
Semester:
first and second
Kind:
mandatory
ECTS:
14
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Content (Syllabus outline)

Metric space and Fourier series:
Definition of metric space, Rn as a metric space, continuinity, sequences and series.
Basic concepts, facts, and techniques in connection with Fourier series, trigonometric series on intervals [-π, π], [-L,L].
Function of several variables, Differential calculus:
Level lines, continuinity, partial derivates and differentiability, Jacobian matrix, implicit function theorem, inverse function theorem, higher derivatives, Taylor formula, applications of differential calculus, extreme, relative extreme.
Space curves and surfaces:
Curves in R3, arc length, tangent, principal normal and binormal, curvature and twist, Frenet’s formulas.
Surfaces in R3. First and second fundamental form, Gauss curtivature.
Integrals dependent on parameters:
Continuinity and differentiability, changing the order of integration, function gamma and beta.
Multiple integration (Riemann integral):
Definitions of double and triple integrals, properties, change of variables, application of double and triple integrals in geometry and physics.
Vector analysis:
Scalar and vector fields, vector differential calculus (grad, div, and curl), line and surface integrals, Gauss theorem, Stokes theorem, Green's formula, and applications in physics.
Analytic functions:
Complex plane, elementary functions in complex, Cauchy-Riemann equations, complex integration, Cauchy theorem and formula, Taylor series, Laurent series, singularities, residue theorem, holomorphic functions as maps, elementary examples.

Readings

M. Dobovišek, Matematika 2, DMFA založništvo, Ljubljana, 2013.
I. Vidav: Višja Matematika I, DMFA založništvo, Ljubljana, 1994, str. 233-329.
I. Vidav: Višja Matematika II, poglavje R. Jamnik: Trigonometrijske vrste, DZS, Ljubljana, 1981, str. 189-221.
I. Vidav: Višja Matematika II, DZS, Ljubljana, 1981, str. 337-381.
I. Vidav: Višja Matematika II, poglavje B. Krušič: Dvojni in mnogoterni integral, DZS, Ljubljana, 1981, str. 299-336.
I. Vidav: Višja Matematika II, poglavje M. Vencelj: Vektorska analiza, DZS, Ljubljana, 1981, str. 383-426.
I. Vidav:Višja Matematika III, poglavje B. Krušič: Kompleksne funkcije, DZS, Ljubljana, 1981, str. 383-426.

Objectives and competences

Students will acquire knowledge about elementary topological property of the space Rn, trigonometric series and their convergence, functions of several variables, differentiability, curves and surfaces in R3, multiple integrals, vector analysis, and fundaments of holomorphic functions. The students will be able to use the acquired knowledge at posing and resolving problems that appears in practices, such as, mechanics, environment sciences, and economics.

Intended learning outcomes

Knowledge and understanding:
Knowledge and understanding of the basic concepts of differential calculus, integration, vector analysis and analytic functions.
Capacity to implement developed methods in geometry and natural sciences.
Application:
Mathematics 2 is one of the basic subjects necessary to understand mechanics and other subjects of natural, technical and social sciences. Knowledge is necessary in modelling of almost all systems.
Reflection:
Integrating theory and practical applications in solving problems.
Transferable skills:
Posing a problem, selection of a method and its application in solving the problem. Analysis of the results from the cases. Skills in using literature. Knowledge is transmitted to virtually all sciences.

Learning and teaching methods

Lectures, exercises, home works, consultations

Assessment

Type (examination, oral, coursework, project):
either 4 midterm exams or written exam at the end of the course
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Barbara Drinovec Drnovšek:
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Holomorphic curves in complex spaces. Duke mathematical journal, ISSN 0012-7094, 2007, vol. 139, no. 2, str. 203-254. [COBISS-SI-ID 14351705]
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. The Poletsky-Rosay theorem on singular complex spaces. Indiana University mathematics journal, ISSN 0022-2518, 2012, vol. 61, no. 4, str. 1407-1423. [COBISS-SI-ID 16679257]
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties. V: Proceedings of Conference on Several Complex Variables on the occasion of Professor Józef Siciak's 80th birthday : July 4-8, 2011, Kraków, Poland, (Annales Polonici Mathematici, ISSN 0066-2216, Vol. 106). Warsaw: Institute of Mathematics, Polish Academy of Sciences, 2012, str. 171-191. [COBISS-SI-ID 16436057]

Jasna Prezelj:
PREZELJ-PERMAN, Jasna. Positivity of metrics on conic neighborhoods of 1-convex submanifolds. International journal of mathematics, ISSN 0129-167X, 2016, vol. 27, no. 5, 1650047 [str. 1-24]. [COBISS-SI-ID 17704537]
PREZELJ-PERMAN, Jasna, SLAPAR, Marko. The generalized Oka-Grauert principle for 1-convex manifolds. Michigan mathematical journal, ISSN 0026-2285, 2011, vol. 60, iss. 3, str. 495-506. [COBISS-SI-ID 16134745]
PREZELJ-PERMAN, Jasna. A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces. Transactions of the American Mathematical Society, ISSN 0002-9947, 2010, vol. 362, no. 8, str. 4213-4228. COBISS-SI-ID 15641433]