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Lecturer(s)
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Šimůnková Martina, RNDr. Ph.D.
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Brzezina Miroslav, doc. RNDr. CSc., dr. h. c.
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Course content
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Lectures: 1. Metric spaces: metric, norm, examples of metric spaces. Inner product, Euclidean space. 2. Metric spaces: other notions (distance of sets, open and close sets, neighborhood of a point, interior points, boundary etc.). 3. Convergence in metric spaces. Mappings between metric spaces, limits and continuity. 4. Completeness, separability, compactness. 5. Functions of several variables, domain, graph, contour lines, level surfaces. 6. Directional and partial derivatives, total differential. Curve, surface, tangent line, normal line, tangent plane. 7. Partial derivatives of the composite function. Implicit function. 8. Local extrema of functions of several variables. 9. Absolut extrema and constrained extrema. 10. The Gauss plane C; convergence in C. Series of complex numbers, the Bolzano-Cauchy condition, absolutely and conditionally convergent series. Criteria of convergence: the comparison test, the ratio (d´Alembert) test, the root (Cauchy) test, the integral test, the Leibniz test. 11. Power series in the complex plane. The radius of convergence, the circle of convergence. Properties of power series. 12. Differentiation and integration of power series. Applications to summation of series. 13. The Taylor series. Trigonometric and exponential functions in the complex plane. 14. Reserve. Exercises are devoted to practise the subject introduced at the last week lecture.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
- Preparation for credit
- 28 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 68 hours per semester
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Learning outcomes
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Metric spaces, functions of several variables, series in the Gauss plane.
Metric spaces, functions of several variables, series in the Gauss plane C.
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Prerequisites
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Calculus 1, Calculus 2, Algebra and geometry 1, Algebra and geometry 2
KMA/KA1 and KMA/KA2
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Assessment methods and criteria
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Oral exam, Written exam
Credit: Active participation on seminars + tests. Exam: writtten and oral
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Recommended literature
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Brabec, J., Hrůza, B.:. Matematická analýza II. Praha, SNTL 1986..
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Černý, I:. Matematická analýza, 2. část. Liberec, TUL 1996..
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Černý, I:. Matematická analýza, 3. část. Liberec, TUL 1996..
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Dont, M. - Opic, B.:. Matematická analýza III - úlohy. Praha, ČVUT 1982..
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Jarník, V.:. Diferenciální počet I. Praha, 1963.
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Jirásek, F. - Čipera, S. - Vacek, M.:. Sbírka řešených příkladů z matematiky II. Praha, SNTL 1989..
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Nekvinda M.:. Matematika II. Liberec, TUL 2000..
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Sikorski, R.:. Diferenciální a integrální počet, Praha, Academia 1973..
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Veselý, J.:. Matematická analýza pro učitele, I, II. Matfyzpress, Praha, 1997..
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