Študijska smer Study field. Samost. delo Individ. work Klinične vaje work. Vaje / Tutorial: Slovensko/Slovene

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1 UČNI NAČRT PREDMETA / COURSE SYLLABUS Predmet: Matematika III Course title: Mathematics III Študijski program in stopnja Study programme and level Univerzitetni študijski program 1.stopnje Fizika First cycle academic study program Physics Študijska smer Study field Letnik Acade mic year Semester Semester vse 2 prvi all 2 first Vrsta predmeta / Course type Univerzitetna koda predmeta / University course code: obvezni predmet/core course??? Predavan ja Lectures Seminar Seminar Vaje Tutorial Klinične vaje work Druge oblike študija Samost. delo Individ. work ECTS Nosilec predmeta / Lecturer: Prof. dr. Miran Černe, prof. dr. Peter Legiša, prof. dr. Bojan Magajna, prof. dr. Janez Mrčun Jeziki / Languages : Predavanja / Slovensko/Slovene Lectures: Vaje / Tutorial: Slovensko/Slovene Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vpis v letnik Opravljen izpit iz vaj je pogoj za pristop k izpitu iz teorije. Med obveznostmi študenta so lahko tudi domače naloge. Prerequisits: Enrollment into the second academic year. The prerequisite for the theoretical exam is a positive result of the written exam. Homework may also be one of obligations. 76

2 Vsebina: Funkciji gama in beta. Dvojni integral, trojni integral in n-terni integral. Prehod na večkratni integral. Težišča, vztrajnostni momenti. Vpeljava novih spremenljivk v večkratni integral, polarne, valjne in krogelne koordinate. Prostor s skalarnim produktom, Hilbertov prostor. Integrali kompleksnih funkcij, prostor L_2[a, b]. Ortonormirani sistemi, Fourierova vrsta, ortonormirana baza. Prostora L_2[-\pi, \pi] in L_2[-a, a], konvergenca trigonometrijske Fourierove vrste po točkah. Navadne diferencialne enačbe (DE): linearna DE prvega reda, ločljive spremenljivke, točne DE. Eksistenčni izrek za DE prvega reda. Eksistenčni izrek za sistem linearnih DE prvega reda. Homogena linearna DE drugega reda: determinanta Wronskega, enačba s konstantnimi koeficienti. Nehomogena linearna DE drugega reda: variacija konstante. Enačba s konstantnimi koeficienti: nastavki. Mehanično (nedušeno in dušeno) nihanje, resonanca. Linearne DE višjih redov. Sistem linearnih DE prvega reda. Sistem s konstantnimi koeficienti: reševanje homogenega sistema z lastnimi Content (Syllabus outline): Gamma and Beta special functions. Double integral, triple integral and multiple integrals. Evaluation by iterated integrals. Center of mass, moment of inertia. Change of variables, polar, cylindric and spherical coordinates. Inner product space, Hilbert space. Integrals of complex functions, space L_2[a,b]. Orthonormal systems, Fourier series, orthonormal basis (complete orthonormal system). Spaces L_2[-\pi,\pi] and L_2[-a,a], pointwise convergence of trigonometric Fourier series. Ordinary differential equations (DE): linear first order DE, separation of variables, exact DE. Existence theorem for the first order DE. Existence theorem for a system of linear first order DE's. Homogeneous linear second order DE: Wronskian, linear DE with constant coefficients. Nonhomogeneous linear second order DE: variation of parameters. DE with constant coefficients: particular solutions. Mechanical (damped and non-damped) oscillations, resonance. Linear DE's of higher order. Systems of linear first order DE's. Systems with with constant coefficients: solving the homogeneus system using eigenvalues and eigenvectors. The exponential function of a square matrix, variation of parameters. 77

3 vrednostmi in lastnimi vektorji. Eksponentna funkcija kvadratne matrike, variacija konstante. Krivuljni integral, potencialna polja, Greenova formula v ravnini. Ploskve v prostoru: normala in tangentna ravnina, površina. Ploskovni integral in pretok vektorskega polja. Ostrogradski-Gaussov in Stokesov izrek, operator nabla. Sestavljene vektorske operacije, Laplaceov operator ( tudi v valjnih in krogelnih koordinatah). Line integrals, potential fields, Green's formula in the plane. Surfaces in R^3, the tangent plane, area of a surface. Surface integrals and the flux of a vector field. Nabla (del) operator, gradient, divergence, curl. Gauss-Ostrogradski and Stokes' theorem. Operations with nabla, Laplacian (also in cylindric and spherical coordinates). Calculus of variations: the Euler equation, isoperimetric problems, extrema with various constraints. Variacijski račun: Eulerjeva DE, izoperimetrični problem, vezani ekstrem. Temeljni literatura in viri / Readings: M. Dobovišek, Nekaj o diferencialnih enačbah, DMFA založništvo, Ljubljana E. Zakrajšek, Analiza III, Matematični rokopisi 21, DMFA-založništvo, Ljubljana 2002 A. Suhadolc, Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, Matematični rokopisi 23, DMFA, Ljubljana, Večina snovi je v (most of the course material is in): W. Kaplan, Advanced Calculus, Addison-Wesley, Boston Pri sestavljanju predavanj so bile uporabljene naslednje knjige (These books were used in compiling the course): M. H. Protter, C. B. Morrey, Intermediate Calculus, 2nd edition, Undergraduate texts in Mathematics, Springer, New York, J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis, Freeman, San Francisco A. Pinkus, S. Zafrany, Fourier Series and Integral Transforms, Cambridge 78

4 University Press, Cambridge G. Bachmann, L. Narici, E. Beckenstein: Fourier and wavelet analysis, Universitext, Springer-Verlag, New York M. Braun, Differential Equations and Their Applications, 4th ed. Applied mathematical sciences 15, Springer-Verlag, New York V. A. Zorich, Mathematical Analysis I and II, Universitext, Springer Verlag, Berlin Heidelberg K. Jaenich, Analysis fuer Physiker und Ingenieure, Funktionentheorie, Differentialgleichungen, Spezielle funktionen, 3. Aufl., Springer Lehrbuch, Springer-Verlag, Berlin Heidelberg L. Elsgolts, Differential equations and the calculus of variations, MIR Publishers, Moscow S. Hassani, Mathematical Physics, A Modern Introduction to its Foundations, Springer-Verlag, New York (V poštev pride le majhen del te obsežne knjige - we need just a fraction of this book.) Priročnik (Handbook): E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, New York 2011 Vaje (problems and solved problems): B. Hvala, Zbirka izpitnih nalog iz analize z namigi, nasveti in rezultati, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, M. Dobovišek, Rešene naloge iz Analize II, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, J. Cimprič, Rešene naloge iz Analize III, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, M. Spiegel: Schaum's Outline of Advanced Mathematics for Engineers and Scientists (Schaum's Outline Series), McGraw-Hill, New York S. Lipschutz, D. Spellman, M. Spiegel: Vector Analysis and an introduction to Tensor Analysis, Second ed. (Schaum's Outline Series), McGraw-Hill, New York

5 Cilji in kompetence: Slušatelj spozna zahtevnejša poglavja matematične analize kot so večkratni integrali, Fourierove vrste, navadne diferencialne enačbe, vektorska analiza, variacijski račun. Matematika 3 je eden osnovnih predmetov pri študiju fizike. Predvideni študijski rezultati: Znanje in razumevanje: Znanje ustreznih definicij in izrekov, razumevanje in deloma repliciranje (vsaj lažjih) dokazov, sposobnost aplikacije pridobljenega znanja, tudi v matematični fiziki. Objectives and competences: Students learn advanced topics in Mathematical Analysis: multiple integrals, Fourier series, ordinary DE, vector analysis, calculus of variations. Mathematics III is a basic course for physicists. Intended learning outcomes: Knowledge and understanding: We expect that students know important definitions and theorems, understand (and ideally be able to replicate) at least the easier proofs, and be able to apply this knowledge, e.g. in Mathematical Physics. Uporaba: Povezava z Matematiko 4, Numeričnimi metodami, Matematično fiziko in drugimi fizikalnimi predmeti. Refleksija: Študent obvlada nekatere zahtevnejše metode matematične analize in jih zna uporabiti v fiziki. Prenosljive spretnosti - niso vezane le na en predmet: Razumevanje uporabnosti splošnejše obravnave matematičnih problemov in višjega nivoja abstrakcije, povezava z že obvladano snovjo. Uporaba domače in tuje literature, reševanje in pravočasno oddajanje domačih nalog. Iskanje podatkov in pomoči v literaturi ali na medmrežju. Študenti si morajo zapomniti važnejše dele snovi. Application: This course is a prerequiste for Mathematics IV, Numerical methods, Mathematical physics, Mechanics and other courses. Reflection: Students master some advanced topics in Mathematical Analysis and are able to apply them in physics. Transferable skills: Students learn to understand the usefulness of the abstract approach, are able to connect the acquired knowledge with what they already mastered. They also learn to use other written sources and the internet. They are able to identify and solve problems, hand in homework on time, and memorize the important topics. 80

6 Metode poučevanja in učenja: Predavanja, vaje, domače vaje, tutorske vaje. Learning and teaching methods: Lectures, tutorials, homework (optional). Načini ocenjevanja: Izpit iz vaj ali dva kolokvija namesto izpita iz vaj, izpit iz teorije, lahko domače naloge. Ocene: 6-10 (pozitivno), 1-5 (negativno) (po Statutu UL). Delež (v %) / Weight (in %) Assessment: Written exam or 2 midterm exams instead of the written exam, oral exam or theoretical test, homework (optional) (pass), 1-5 (fail) (according to the Statute of UL) Reference nosilca / Lecturer's references: Prof. dr. Miran Černe: - M. Černe, M. Zajec, Boundary differential relations for holomorphic functions on the disc. Proc. Am. Math. Soc. 139 (2011), M. Černe, M. Flores, Generalized Ahlfors functions. Trans. Am. Math. Soc. 359 (2007), M. Černe, M. Flores, Quasilinear -equation on bordered Riemann surfaces. Math. Ann. 335 (2006), prof. dr. P. Legiša: - P. Legiša, Adjacency preserving mappings on real symmetric matrices. Math. commun., Croat. Math. Soc., Divis. Osijek, 2011, vol. 16, no. 2, P. Legiša, Automorphisms of M n, partially ordered by the star order. Linear multilinear algebra, 2006, vol. 54, no. 3, P. Legiša, Automorphisms of M n, partially ordered by rank subtractivity ordering. Linear algebra appl. 2004, vol. 389, prof. dr. B. Magajna: - B. Magajna, Sums of products of positive operators and spectra of Lüders operators, Proc. Amer. Math. Soc. 141 (2013) B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math. Anal. Appl. 389 (2012)

7 - B. Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal. 129 (1995) prof. dr. J. Mrčun: - I. Moerdijk, J. Mrčun: On the developability of Lie subalgebroids. Adv. Math. 210 (2007), J. Mrčun: On isomorphisms of algebras of smooth functions. Proc. Amer. Math. Soc. 133 (2005), I. Moerdijk, J. Mrčun: On integrability of infinitesimal actions. Amer. J. Math. 124 (2002),

Študijska smer Study field. Samost. delo Individ. work Klinične vaje work. Vaje / Tutorial: Slovensko/Slovene

Študijska smer Study field. Samost. delo Individ. work Klinične vaje work. Vaje / Tutorial: Slovensko/Slovene UČNI NAČRT PREDMETA / COURSE SYLLABUS Predmet: Matematika 2 Course title: Mathematics 2 Študijski program in stopnja Study programme and level Univerzitetni študijski program 1.stopnje Fizika First cycle