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Mathematics 2

Code: 38183
ECTS: 8.0
Lecturers in charge: doc. dr. sc. Franka Miriam Brückler - Lectures
Take exam: Studomat
Load:

1. komponenta

Lecture typeTotal
Lectures 60
Exercises 45
* Load is given in academic hour (1 academic hour = 45 minutes)
Description:
1. Systems of linear equations, the Gauss algorithm, determinants, Cramer's rule. (14 + 0 + 11 hours)
2. Basic linear algebra: vectors, basis, coordinatisation (continued from Mathematics 1). (6 + 0 + 3 hours)
3. Multivariable functions. Partial derivatives (definition and geometric interpretation). (3 + 0 + 2 hours)
4. Gradient. Directional derivative. Higher-order partial derivatives. (2 + 0 + 1 hours)
5. Implicitely represented functions. Surfaces in space (equation of the tangential plane and normal line). Curves in space (equation of the tangential line and normal plane). (3 + 0 + 2 hours)
6. Analysing of two-variable functions. Extrema of two- and three-variable functions. (1 + 0 + 3 hours)
7. Conditional extrema. Lagrange multipliers. Applications to optimization problems. (2 + 0 + 2 hours)
8. Riemann integrals of two- and three-variable functions. Fubini theorem for two- and three-variable functions. Change of variables in double and triple integrals. (2 + 0 + 1 hours)
9. Integration in polar, cylindric and spheric coordinates. Applications of double and triple integrals (barycenter, moment of inertia). (2 + 0 + 2 hours)
10. Line integrals of first and second sort. Vector fields. Rotation and divergence. Conservative vector fields. Examples from mechanics. (5 + 0 + 2 hours)
11. The notion of a differential equation. Types of differential equations. The order of a differential equation. Linear ordinary differential equations of first order (definition and solution). (3 + 0 + 2 hours)
12. Nonlinear first order ordinary differential equations (Bernoulli equation, Ricatti equatios, separable equations, logistic equation with applications, exact equations, Euler multiplier). (3 + 0 + 5 hours)
13. Linear second order ordinary differential equations reda (fundamental set, Wronskian, method of indeterminate coefficients for determining a particular solution, method of variation of constants). (4 + 0 + 4 hours)
14. Sequences and series: Definition and basic properties, convergence, convergence criteria. (3 + 0 + 2 hours)
15. Applications of higher-order derivatives to approximations of functions with polynomials. Approximation with Taylor polynomials. Approximate calculations and estimate of the error of approximatin by a Taylor polynomial of degree n. Taylor series. (3 + 0 + 2 hours)
16. Fourier series (definition and basic properties). (4 + 0 + 1 hours)


LEARNING OUTCOMES:
- to solve systems of linear equations with various methods
- to explain and to understand basic matrix algebra and basic knowledge about abstract vector spaces
- to explain and to understand fundamentals of multivariable calculus
- to explain and to understand advanced multivariable calculus notions - gradient, divergence, rotation, differentials, line integrals
- to explain and to understand the notion of ordinary differential equations and ability to solve basic types of ordinary differential equations
- to explain and to understand the definitions and basic properties of sequences, series and function series
- to explain and to understand basics of Fourier analysis
Literature:
  1. M. Alić: Obične diferencijalne jednadžbe, PMF-Matematički odjel, Zagreb, 1994.
  2. B. P. Demidovič: Zadaci i riješeni primjeri iz više matematike, Tehnička knjiga, Zagreb, 1978.
  3. S. Kurepa: Matematička analiza III, Tehnička knjiga, Zagreb, 1975.
  4. F. Ayres, E.Mendelson: Differential and Integral Calculus, Schaum's Outline Series, New York, 1990.
Prerequisit for:
Enrollment :
Attended : Mathematics 1
2. semester
Mandatory course - Mandatory studij - Chemistry
Consultations schedule: