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Lecturer(s)
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Švrček Jaroslav, RNDr. CSc.
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Calábek Pavel, RNDr. Ph.D.
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Course content
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1. Sequences and series of functions: Pointwise and uniform convergence, convergence criteria (esp. the Weierstrass criterion). Properties of the limit function - limit, continuity, derivative and integral. 2. Power series: Radius, interval and domain of convergence. Uniform convergence of power series. Taylor series, Taylor expansion of elementary functions. Approximate computing via series. 3. Metric spaces: Metric on a set, examples of metric spaces. Normed linear space. Classification of points according to a set. Open and closed sets and their properties. Convergent and Cauchy sequences of points. 4. Functions and mappings in Euclid spaces: Practical aplications. Limit and continuity of a mapping (function). Properties of continuous functions on compact sets 5. Differential calculus in R^n: Partial derivatives and directional derivatives in R^n. Partial derivatives of higher order, interchanging the order of differentiation, total differential of a function and its application in approximate computing. Partial derivatives of compound functions. Differentials of higher order. The Taylor formula. Local extrema of functions, global extrema. 6. Implicit functions: Implicit functions of a single variable, its existence, uniqueness and differentiability. Extrema of implicit functions. Implicit functions of several variables. Constraint extrema, method of the Lagrange multipliers. 7. Integral calculus in R^n: The Jordan measure of a set in R^n. Properties of the measure. Definition and fundamental properties of the Riemann integral in R^n, its geometric interpretation. Multiple integration over intervals and normal domains. Substitution in integrals, especially polar, cylindrical and spherical coordinates. Practical aplications.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Understand basic notions concerning function series and metric spaces. Summary of topics of mathematical analysis.
Comprehension Understand basic notions concerning function series and metric spaces. Describe basic methods of mathematical analysis
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Prerequisites
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unspecified
KAG/KMAI
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Assessment methods and criteria
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Oral exam, Written exam
Credit: made homework. Exam: pass written test and the student has to understand the subject and prove principal results.
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Recommended literature
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Brabec,J., Hrůza, B. (1989). Matematická analýza II. SNTL, Praha.
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Braun M. (1983). Differential equations and their applications. Springer-Verlag New York.
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Gillman L., MCDOWELL R. H. (1973). Calculus. W. W. Norton & Company Inc. New York.
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Kojecká J., Rachůnková I. (1989). Řešené příkklady z matematické analýzy III.. Olomouc.
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Novák V. (1985). Nekonečné řady. UJEP Brno.
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Trávníček S. (2006). Matematická analýza I a III (učební text na internetu). KAG PřF UP Olomouc.
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V. Jarník. (1976). Diferenciální počet I a II. SPN, Praha.
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