Mathematics II

Enrollment into the program

Fourier 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 continuinity, sequences and series in Rn 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. Systems of ODEs: Systems of 1st order ODEs, existence theorem, solutions to the homogeneous and nonhomogeneous constant coefficient systems, phase space, analysis of stationary points, stability

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.

Students will acquire knowledge about 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.

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.
 

Lectures, exercises, homeworks, consultations

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)

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. 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]

– 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]