# Mathematics

Bachelor's programme in Environment (first cycle)

## Objectives and competences

Meet and complement the basic concepts and principles of higher mathematics and their applications in science and technology. Developing thinking skills specific to problem solving: understanding the problem, finding appropriate data rescue plan, rescue, checking compliance solutions and analysis solution. Expression of the mathematical reasoning and use it in the science and technical research.

## Prerequisites

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## Content

h1. 1. Sets and Numbers

1. Sets

2. Natural, integers, rational, real numbers

3. Operations with real numbers

4. Complex numbers

5. Induction

h1. 2. Linear algebra

1. Systems, Gauss elimination

2. Matrices

3. Determinants

4. Vectors; algebraic and geometrical properties of vectors

5. Scalar, cross and triple products

6. Eigenvalues and eigenvectors

7. Equation of a line and of a plane in the space

h1. 3. Functions, Limits, and Continuity

1. Fundamental concepts: bounded functions, monotonic functions, injections, surjections, bijections

2. Composition and invertible functions

3. Elementary functions

4. Limits

5. Continuity

h1. 4. Derivatives

1. Definition of a derivative

2. Derivatives of elementary functions

3. Higher derivatives

4. Relative extrema

5. Convexity and points of inflection

6. Graph of a function

7. L'Hopital's rule

h1. 5. Integrals

1. Indefinite and definite integrals

2. Special methods of integration (by parts and by substitution)

h1. 6. Application of integrals

1. Arc length. Area. Volumes

2. Applications in physics

h1. 7. Functions in more variables

1. Graphs, level curves

2. Partial derivatives, gradient, divergence

3. Relative extrema

## Intended learning outcomes

By the end of this course students will be able to:

• do operations with matrices,

• calculate determinants,

• solve linear systems,

• find the eigenvalues and eigenvectors of a matrix,

• solve problems with vectors, lines and planes in the space,

• calculate limits, derivatives, integrals,

• draw the graph of a function,

• calculate partial derivatives and relative extrema of functions with more variables.

## Readings

- G. Tomšič, B.Orel, N. Mramor Kosta, Matematika I, Založba FE in FRI, Ljubljana, 2000 (3.izdaja).
- B. Orel, Linearna algebra, 2020.
**E-version** - M. Zeljko, Matematične metode.
**E-version** - P. Oblak, Matematika.
**E-version** - R. Courant, J. Fritz, Introduction to Calculus and Analysis, vol.1, Springer, 1999.
**Catalogue** - David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron, Linear Algebra, 2013.
**E-version** - W. Trench, Introduction to Real Mathematics, 2013.
**E-version** - Slovensko angleški matematični slovar.
**E-version**

## Assessment

Completed homework (20 %), written examination (80 %)

## Lecturer's references

Associate Professor of Mathematics at the University of Nova Gorica

[1.] CRISTEA, Irina. Regularity of intuitionistic fuzzy relations on hypergroupoids. V: Worksop on a new approach in theoretical and applied methods in algebra and analysis, Constanta, Romania, April 4-6, 2013 [...] and Workshop on Algebraic and Analytic Number Theory and their Applications, Constanta, Romania, May 23-24, 2013, (Analele Universitǎţii "Ovidius" Constanţa, ISSN 1224-1784, vol. 22, no. 1). Constanţa: Ovidius University Press, 2014, vol. 22, no. 1, str. 105-119, doi: 10.2478/auom-2014-0009. [COBISS.SI-ID 3277563]

[2.] CRISTEA, Irina, JANČIĆ-RAŠOVIĆ, Sanja. Composition hyperrings. An. Univ. "Ovidius" Constanţa, Ser. Mat. (Print), 2013, vol. 21, no. 2, str. 81-94. [COBISS.SI-ID 2850555]

[3.] ANGHELUŢǍ, Carmen, CRISTEA, Irina. On Atanassov's intuitionistic fuzzy grade of complete hypergroups. J. mult.-valued log. soft comput.. [Print ed.], 2013, vol. 20, no. 1/2, str. 55-74. [COBISS.SI-ID 2623995]

[4.] CRISTEA, Irina, ZHAN, Jianming. Lower and upper fuzzy topological subhypergroups. Acta math. Sin., Engl. ser. (Print), 2013, vol. 29, no. 2, str. 315-330, [COBISS.SI-ID 2634747]

[5.] CRISTEA, Irina, ŞTEFǍNESCU, Mirela, ANGHELUŢǍ, Carmen. About the fundamental relations defined on the hypergroupoids associated with binary relations. Eur. j. comb., 2011, vol. 32, no. 1, str. 72-81. [COBISS.SI-ID 2062587