Mathematics 1

Enrollment into the program.

Positive result of the written exam is a prerequisite for the oral/theoretical exam.

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.

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.

F. Križanič, Temelji realne matematične analize. Državna založba Slovenije, Ljubljana, 1990.
M. H. Protter, C. B. Morrey, Intermediate Calculus. Springer-Verlag, New York-Heidelberg, 1985.
W. Rudin, Principles of mathematical analysis. McGraw-Hill, Auckland, 1976.
I. Vidav, Višja matematika I. Društvo matematikov, fizikov in astronomov Slovenije, Ljubljana, 1994.

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.

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.

Transferable skills: Skills in using the literature and other sources, the ability to identify and solve the problem, critical analysis.

Lectures and tutorial sessions, homework.

2 midterm exams instead of written exam, written exam, homework (optional)
Oral exam / theoretical test.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

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, 419-432.

• P. Legiša, Automorphisms of Mn, partially ordered by the star order. Linear
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  multilinear algebra, 2006, vol. 54, no. 3, 157-188.

• P. Legiša, Automorphisms of Mn, partially ordered by rank subtractivity
  ordering. Linear algebra appl. 2004, vol. 389, 147-158.
  prof. dr. B. Magajna:

• B. Magajna, Sums of products of positive operators and spectra of Lüders
  operators, Proc. Amer. Math. Soc. 141 (2013) 1349-1360.

• B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math.
  Anal. Appl. 389 (2012) 1291-1302.

• B. Magajna, The Haagerup norm on the tensor product of operator modules,
  J. Funct. Anal. 129 (1995) 325-348.
  prof. dr. J. Mrčun:

• I. Moerdijk, J. Mrčun: On the developability of Lie subalgebroids. Adv. Math.
  210 (2007), 1-21.

• J. Mrčun: On isomorphisms of algebras of smooth functions. Proc. Amer. Math.
  Soc. 133 (2005), 3109-3113.

• I. Moerdijk, J. Mrčun: On integrability of infinitesimal actions. Amer. J. Math.
  124 (2002), 567-593.
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