Mathematics II

2021/2022
Programme:
Applied Physisc, First Cycle
Year:
year
Semester:
first and second
Kind:
mandatory
ECTS:
16
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Enrollment into the program

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.

Differential equations:

Ordinary differential equations:
Separable differential equation, first order linear differential equation,
Euler differential equation, Bernoulli differential equation, Ricatti differential equation, exact differential equation, existence and uniqueness of solutions.

Higher order linear differential equation:
Homogeneous equation, Wronskian, nonhomogeneuous equation, method of undetermined coefficients, method of variation of constants.
Oscillations, reverse loop control

Readings

M. Dobovišek, Matematika 2, DMFA založništvo, Ljubljana, 2013.
M. Dobovišek, Nekaj o diferencialnih enačbah, DMFA založništvo, 2011.
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.

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 some first and second order diferential equations.The students will be able to use the acquired knowledge at posing and resolving problems that appears in practics, 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, homeworks, consultations

Assessment

Either 4 midterm exams or written exam at the end of the course
Oral exam
Two mark at this subject.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

[1] DOBOVIŠEK, Mirko, KUZMA, Bojan, LEŠNJAK, Gorazd, LI, Chi-Kwong, PETEK, Tatjana. Mappings that preserve pairs of operators with zero triple Jordan product. Linear algebra appl.. [Print ed.], 2007, vol. 426, iss. 2-3, str. 255-279.
[2] DOBOVIŠEK, Mirko. Maps from M [sub] n(F) to F that are multiplicative with respect to the Jordan triple product. Publ. math. (Debr.), 2008, vol. 73, fasc. 1-2, str. 89-100.
[3] DOBOVIŠEK, Mirko, Nekaj o diferencialnih enačbah, (Izbrana poglavja iz matematike in računalništva, 47). 1. natis. Ljubljana: DMFA - založništvo, 2011. 131 str.
[4] DOBOVIŠEK, Mirko, Maps from M[sub]2 to M[sub]3 that are multiplicative with respect to the Jordan triple product. Aequationes Mathematicae, Vol. 85(2013), 539-552.
[5] DOBOVIŠEK, Mirko, Matematika 2, (Izbrana poglavja iz matematike in računalništva, 48). 1, natis. Ljubljana: DMFA - založništvo, 2013, 340 str.