Analysis 2a

2022/2023
Programme:
Mathematics, First Cycle
Year:
2 year
Semester:
first
Kind:
mandatory
ECTS:
8
Language:
slovenian
Hours per week – 1. semester:
Lectures
4
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Completed course Analysis 1.

Content (Syllabus outline)

Functions of several variables. Continuity. Partial derivatives and differentiability. Jacobian matrix and determinant. Differential of a composed mapping. Implicit function theorem and inverse function theorem. Higher order partial derivatives. Taylor's formula. Free and constrained extrema.
Parametric integrals, continuity and differentiability. Fubini's theorem.
Curves and surfaces in space. Submanifolds.
Multiple Riemann integral. Properties. Sets with zero volume and measure. Theorem on the existence of integrals. Transformation into a multiple integral. Improper integrals. The use of integrals in geometry and physics.

Readings

Vidav: Višja Matematika I, DMFA-založništvo, Ljubljana, 1994.
Vidav: Višja Matematika II, DZS, Ljubljana, 1981.
T. M. Apostol: Calculus II : Multi-Variable Calculus and Linear Algebra with Applications, 2nd edition, John Wiley & Sons, New York, 1975.
J. E. Marsden, A. J. Tromba: Vector Calculus, 5th edition, Freeman, New York, 2004.
Suhadolc: Integralske transformacije/Integralske enačbe, DMFA-založništvo, Ljubljana, 1994.
A. Suhadolc: Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, DMFA-založništvo, 1998

Objectives and competences

Student becomes familiar with the differential and the integral calculus of functions of several real variables.

Intended learning outcomes

Knowledge and understanding: Undestanding of the differential and the integral calculus of functions of several variables and related topics. Application of these methods in geometry and natural science.
Application: Analysis 2a is one of the fundamental courses in mathematical studies. It is a prerequisite for the courses Analysis 3, Measure theory, Functional analysis, Probability and statistics, Analysis on manifolds.
Reflection: Understanding of the theory based on examples and 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 domestic and foreign literature.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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, FORSTNERIČ, Franc. Embedding some bordered Riemann surfaces in the affine plane. Mathematical research letters, ISSN 1073-2780, 2002, vol. 9, no. 5-6, str. 683-696. [COBISS-SI-ID 12391257]
ČERNE, Miran. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. Arkiv för matematik, ISSN 0004-2080, 2002, vol. 40, no. 1, str. 27-45. [COBISS-SI-ID 11623513]
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. Proper discs in Stein manifolds avoiding complete pluripolar sets. Mathematical research letters, ISSN 1073-2780, 2004, vol. 11, no. 5-6, str. 575-581. [COBISS-SI-ID 13311065]
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]
FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]
FORSTNERIČ, Franc, ROSAY, Jean-Pierre. Approximation of biholomorphic mappings by automorphisms of C [sup] n. Inventiones Mathematicae, ISSN 0020-9910, 1993, let. 112, št. 2, str. 323-349. [COBISS-SI-ID 9452121]