Študijska smer Study field. Klinične vaje work. Nosilec predmeta / prof. dr. Peter Legiša, prof. dr. Bojan Magajna, prof. dr.
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1 UČNI NAČRT PREDMETA / COURSE SYLLABUS Predmet: Matematika 1 Course title: Mathematics 1 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 1 prvi all 1 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. 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 ustnemu/teoretičnemu izpitu. Prerequisits: Enrollment into the program. Positive result of the written exam is a prerequisite for the oral/theoretical exam. 38
2 Vsebina: Množice: družine množic, kartezični produkt. Relacija, ekvivalenčna relacija. Funkcije: injektivna, surjektivna, bijektivna funkcija, graf funkcije, inverzna funkcija, kompozitum funkcij. Naravna števila, princip popolne indukcije. Cela, racionalna števila. Realna števila: urejenost realnih števil, supremum, infimum, polnost. Kompleksna števila: polarni zapis, koreni enote. Vektorji v trirazsežnem prostoru. Skalarni, vektorski, mešani produkt, dvojni vektorski produkt. Vzporedni premik koordinatnega sistema, daljica, konveksnost. Enačba ravnine, premice. Zaporedje realnih ali kompleksnih števil: stekališče, limita. Cauchyjevo zaporedje. Vsako omejeno zaporedje realnih ali kompleksnih števil ima stekališče. Računanje z zaporedji. Število e. Realne funkcije realne spremenljivke. Zveznost funkcije. Računanje z zveznimi realnimi funkcijami. Lastnosti realnih zveznih funkcij. Enakomerna zveznost. Limita funkcije. Inverzna funkcija k monotoni realni funkciji. Krožne funkcije, eksponentna funkcija, logaritemska funkcija. Limite v neskončnosti in neskončne limite, asimptote grafov. Parametrično podane krivulje. Content (Syllabus outline): Sets: families of sets, cartesian product. Relation, equivalence relation. Functions: injective, surjective, bijective function, graph of a function, inverse function, composition of functions. Natural numbers, mathematical induction. Integers, rational numbers. Real numbers: ordering real numbers, supremum, infimum, completeness. Complex numbers: polar form, roots of unity. Vectors in the real three-dimensional space. Inner, cross, scalar-valued triple product, vector-valued triple product. Translation of a coordinate system, line segment, convexity. Equation of a plane, line. Sequences of real and complex numbers: accumulation point, limit point. Cauchy sequences. Any bounded sequence of real or complex numbers has an accumulation point. Arithmetic operations on sequences. Number e. Real functions of one real variable. Continuity of functions. Computing with continuous real functions. Properties of continuous real functions. Uniform continuity. Limit value of a function. Inverse function of a monotone function. Cyclometric functions, exponential function, logarithmic function. Limit values in infinity and infinite limit values, asymptotes of graphs. Parametric curves. 39
3 Odvod. Aproksimacija z odvodom. Pravila za odvajanje. Odvodi elementarnih funkcij. Višji odvodi. Lokalni ekstremi, Rollov in Lagrangeov izrek. L' Hospitalova pravila. Risanje funkcij. Newtonova metoda iskanja ničel. Nedoločeni integral. Določeni integral: spodnje in zgornje vsote, Riemannove vsote. Lastnosti integrala. Glavni izrek integralskega računa. Povprečna vrednost. (Numerična integracija: trapezna in Simpsonova formula.) Izlimitirani integrali. Ploščine, prostornine in površine rotacijskih teles. Krivulje v trirazsežnem prostoru: dolžina loka, ukrivljenost, torzija, Frenet Serretjeve formule. Vrste realnih in kompleksnih števil. Absolutna konvergenca. Konvergenčni kriteriji: kvocientni, korenski, integralski. Funkcijska zaporedja in vrste, enakomerna konvergenca. Odvajanje in integriranje funkcijskih zaporedij in vrst. Potenčne vrste, konvergenčni radij. Taylorjeva formula in Taylorjeva vrsta. Eksponentna vrsta, logaritemska vrsta, binomska vrsta, vrsti za funkciji sinus in kosinus. Eksponentna funkcija v kompleksnem. Večrazsežni evklidski prostor. Funkcije več spremenljivk. Zvezne funkcije več spremenljivk in njihove lastnosti. Parcialni odvodi, višji parcialni odvodi. Taylorjeva vrsta funkcije več spremenljivk. Ekstremi funkcije več spremenljivk. Zadostni pogoj za ekstrem funkcije dveh spremenljivk. Integrali s parametrom. Derivative. Approximation with the derivative. Differentiation rules. Derivatives of elementary functions. Higher derivatives. Local extrema, Rolle's theorem, Langrage's mean value theorem. L'Hospital's rules. Drawing graphs of functions. Newton's method for finding zeros of functions. Indefinite integral. Definite integral: upper sums, lower sums, Riemann sums. Properties of the integral. The main theorem of integral calculus. Average value. (Numerical integration: trapezoidal and Simpson's rule). Improper integrals. Areas, volume and area of a surface of revolution. Curves in three-dimensional space: arc length, curvature, torsion, Frenet Serret formulas. Series of real and complex numbers. Absolute convergence. Convergence tests: ratio test, root test, integral test. Sequences and series of functions, uniform convergence. Differentiation and integration of sequences and series of functions. Power series, radius of convergence. Taylor formula and Taylor series. Exponential series, logarithmic series, binomial series, series for sine and cosine. Exponential series in complex numbers. Higher dimensional Euclidean space. Functions of several variables. Continuous functions of several variables and their properties. Partial derivatives, higher partial derivatives. Taylor series of a function of several variables. Extrema of a function of several variables. Sufficient condition for an extremum of a function of two variables. Parameter-dependent integrals. 40
4 Temeljni literatura in viri / Readings: 1. F. Križanič, Temelji realne matematične analize. Državna založba Slovenije, Ljubljana, M. H. Protter, C. B. Morrey, Intermediate Calculus. Springer-Verlag, New York-Heidelberg, W. Rudin, Principles of mathematical analysis. McGraw-Hill, Auckland, I. Vidav, Višja matematika I. Društvo matematikov, fizikov in astronomov Slovenije, Ljubljana, Cilji in kompetence: Študent spozna osnovne pojme matematične analize, kot so limita, zveznost, odvod in integral funkcije ene realne spremenljivke, številske in funkcijske vrste, poleg tega pa tudi zveznost ter odvod funkcije več realnih spremenljivk. Matematika 1 sodi med temeljne predmete pri študiju fizike. Predvideni študijski rezultati: Znanje in razumevanje: Poznavanje in razumevanje osnovnih pojmov, definicij in izrekov. Uporaba: Matematika 1 sodi med temeljne predmete pri študiju fizike. Razumevanje snovi predmeta je nepogrešljivo pri mnogih drugih matematičnih in fizikalnih predmetih na programu ter pri reševanju problemov v fiziki. Refleksija: Razumevanje teorije na podlagi uporabe. Objectives and competences: Student learns the basic concepts of mathematical analysis such as limit, continuity, derivative and integral of real functions of one real variable, numerical and function series, and continuity and differentiation of real functions of several real variables. Mathematics 1 is one of the fundamental courses of the study of physics. Intended learning outcomes: Knowledge and understanding: Knowledge and understanding of basic notions, definitions and theorems. Application: Mathematics 1 is one of the fundamental courses of the study of physics. Understanding the material of this course is indispensable for many other mathematics and physics courses of the program and for solving problems in physics. Reflection: Understanding the theory from the applications. 41
5 Prenosljive spretnosti - niso vezane le na en predmet: Spretnosti uporabe domače in tuje literature in drugih virov, identifikacija in reševanje problemov, kritična analiza. Transferable skills: Skills in using the literature and other sources, the ability to identify and solve the problem, critical analysis. Metode poučevanja in učenja: Predavanja in vaje, domače naloge. Learning and teaching methods: Lectures and tutorial sessions, homework. Načini ocenjevanja: 2 kolokvija namesto izpita iz vaj, izpit iz vaj, domače naloge (neobvezno), ustni izpit / izpit iz teorije (pozitivno), in 1-5 (negativno) (po Statutu UL). Delež (v %) / Weight (in %) Assessment: 2 midterm exams instead of written exam, written exam, homework (optional), oral exam / theoretical test (pass), 1-5 (fail) (according to the Statute of UL) Reference nosilca / Lecturer's references: 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 42
6 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) 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),
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