Analysis I

The primary goal of this course is to give students the knowledge of methods of classic mathematical analysis: calculation of limits at various conditions, calculation of derivatives of functions with one and many argument, calculation of integrals. Basic goal is following: to give the knowledge for analysis of functions.

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1.
Introduction to analysis
(functions; limits; infinitely small and large quantities; continuity of functions)
2.
Differentiation of functions (calculating derivatives directly; tabular
differentiation; derivatives of implicit functions; derivatives of higher orders; differentials of first and higher orders; mean value theorems; Taylor formula; L'Hospital-Bernoulli rule for evaluating indeterminate forms.)
3.
Extrema of a function and geometric applications of a derivative
(extrema of one argument functions, direction of concavity; points of inflection; asymptotes; graphing functions by characteristic points; differential of an arc curvature)
4.
Indefinite integrals
(direct integration; integration by substitution; integration by parts; standard integrals containing a quadratic trinomial; integration of rational functions; integration of irrational functions; integrating trigonometric functions; integration of hyperbolic functions; integration using trigonometric and hyperbolic substitutions; integration of various transcendental functions; reduction formulas.)
5.
Definite integrals
(definite integral as a limit of a sum; evaluating definite integrals by means of indefinite integrals; improper integrals; change of variable in a definite integral; integration by parts; areas of plane figures; the arc length of a curve; volumes of solids; applying definite integrals to the solution of physical problems.)
6.
Series
(number series; functional series; Taylor's series; Fourier's series.)
7.
Functions of several variables
(concepts; continuity; partial derivatives; total differential of a function; differentiation of composite functions; derivative in a given direction and the gradient of a function; higher- order derivatives and differentials; integration of total differentials; differentiation of implicit functions; change of variables; tangent plane and the normal to a surface; Taylor’s formula for a function of several variables; extreme of a function of several variables; largest and smallest values of functions; singular points of plane curves; envelope; arc length of a space curve; vector function of a scalar argument; natural trihedron of a space curve; curvature and torsion of a space curve.)
8.
Multiple and line integrals
(double integral in rectangular coordinates; change of variables in a double integral; computing areas and volumes; double and triple integrals; triple integral in cylindrical and spherical coordinates; line and surface integrals; Ostrogradsky-Gauss Formula; Stokes.)

Students will get the information about finding of extreme points of functions, about asymptotic behaviour of functions, about roots of functions etc. Finally, they will have knowledge for total analysis of functions.

• R. Courant, Differential and Integral Calculus I. Blackie & Son Ltd (1961). E-version
• B. Demidovich et al., Problems in Mathematical Analysis, Mir Publishers (1972). E-version

• written tests, writen exam
• oral exam

Doc. Dr. Hashem Bordbar je habilitiran za področje matematike na Univrzi v Novi Gorici. Assistant Professor Hashem Bordbar has habiliation in the field of mathematics at the University of Nova Gorica
Hashem Bordbar, "Ordered algebraic structure in a linear code", Advances in mathematics of communications, 2025, vol. 19, issue 4, str. 1248-1259, DOI: 10.3934/amc.2024052.
Hashem Bordbar, "Torsion elements and torsionable hypermodules", Mathematics, 2023, vol. 11, issue 21, str. 4525, DOI: 10.3390/math11214525.
Hashem Bordbar, "BCK codes", Advances in computational intelligence, Feb. 2022, vol. 2, iss. 1, str. 1-9, DOI: 10.1007/s43674-021-00018-4.
Arezoo Soufi Karbaski, Taher Abualrub, Irfan Siap, Hashem Bordbar, "On the construction of nonbinary LCD quadratic residue and double quadratic residue codes", Finite fields and their applications, Mar. 2025, vol. 103, str.102591, DOI: 10.1016/j.ffa.2025.102591
Hashem Bordbar, Ravi Kumar Bandaru, Young Bae Jun, "Soft GE-algebras", Afrika matematika, Jun. 2025, vol. 36, str. 125, DOI: 10.1007/s13370-025-01347-0.