Analysis 3

2022/2023
Programme:
Mathematics Education
Year:
3 year
Semester:
first
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Completed courses Analysis 1, Analysis 2a.

Content (Syllabus outline)

Introduction: first-order differential equations, phase space, vector fields, integral curves. Examples of the first order differential equations. Exact equation and integrating factors. Singular solutions and parametric solutions. Clairot equation. Examples. Existence uniqueness theorem and dependence of the solutions on the initial conditions. General solution. Flow of the ordinary differential equation. First-order system of differential equations. Linearization. Flow of a homogenuous system of linear differential equations. Liouville formula. System with constant coefficients. Higher order linear
differential equations with constant coefficients. Forced and damped oscillations. Singular points of a linear system. Stability.
Calculus of variations. Some examples. Banach space and linear operators on Banach spaces.
Derivatives of the operators on Banach spaces. Extremals. Fundamental theorem of calculus of variations. Euler-Lagrange equation and its solution. Legendre transformation and the canonical system. Variable end-point problems.
Isoparametric problem. Lagrange problem.

Readings

F. Križanič: Navadne diferencialne enačbe in variacijski račun, DZS, Ljubljana, 1974.
E. Zakrajšek: Analiza III, DMFA-založništvo, Ljubljana, 2002.
V. I. Arnold: Ordinary Differential Equations, MIT Press, Cambridge, 1978.
W. Walter: Ordinary Differential Equations, Springer, New York, 1998.
S. Lefschetz: Differential Equations : Geometric Theory, 2nd edition, Dover Publications, New York, 1977.
L. Perko: Differential Equations and Dynamical Systems, 3rd edition, Springer, New York, 2004.

Objectives and competences

The student is introduced to the concept of differential equations and their solutions. Learning to solve or treat certain types of ordinary differential equations with special emphasis on linear equations. They learn the basics of calculus of variations and with the concept of the derivative of the operator between Banach spaces.

Intended learning outcomes

Knowledge and understanding:
Understanding the concept of differential equations and their solutions. Handling analytical procedures for solving certain types of differential equations. Understanding the concept of variational calculus.
Application: The formulation of some problems in mathematics, natural sciences and social sciences in the form of differential equations and solving them. Formulation of some mathematical and physical problems in the form of variational calculus and solving them.
Reflection: Understanding of the theory from the applications.
Transferable skills: Identifying and solving problems. Formulation of some non mathematical problems in mathematical language. Ability to use domestic and foreign literature.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Type (examination, oral, coursework, project):
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

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]
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, FORSTNERIČ, Franc. Approximation of holomorphic mappings on strongly pseudoconvex domains. Forum mathematicum, ISSN 0933-7741, 2008, vol. 20, iss. 5, str. 817-840. [COBISS-SI-ID 15078745]
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]
Pavle Saksida:
SAKSIDA, Pavle. On zero-curvature condition and Fourier analysis. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2011, vol. 44, no. 8, 085203 (19 str.). [COBISS-SI-ID 15909465]
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]
SAKSIDA, Pavle. Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theory. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2005, vol. 38, no. 48, str. 10321-10344. [COBISS-SI-ID 13802073]