| Course title | Mathematical Analysis 3 |
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| Course code | KMA/KAN3 |
| Organizational form of instruction | Seminary |
| Level of course | Bachelor |
| Year of study | 2 |
| Semester | Winter |
| Number of ECTS credits | 5 |
| Language of instruction | Czech |
| Status of course | Compulsory |
| Form of instruction | Face-to-face |
| Work placements | Course does not contain work placement |
| Recommended optional programme components | None |
| Course availability | The course is available to visiting students |
| Lecturer(s) |
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| Course content |
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Course Description The content of the course is mastering the basics of differential calculus of functions of several real variables and the theory of series of functions. Functions of Several Variables Functions of several variables, graph of a function of several variables, contour lines. The concept of a neighborhood in a multidimensional space, continuity and limit of functions of several variables. Differential Calculus of Several Variables Differentiability of a function of several variables, total differential, necessary condition for differentiability, sufficient condition for differentiability, partial derivatives, gradient, equation of a tangent plane. Directional derivatives. The technique of differentiating composite functions, connection with knowledge from algebra. Extrema and Metric Spaces The concept of continuity, limit, and extremum with respect to a set. Free and constrained extrema of a function of several variables. Local and absolute (global) extrema. Necessary condition for the existence of a local extremum, sufficient condition for the existence of a local extremum. The method of Lagrange multipliers. Basic concepts of the theory of metric spaces and their application for functions of several variables. Completeness and compactness of a metric space, Weierstrass theorem. Double Integrals Double Riemann integral on a rectangle, iterated integral, Fubini's theorem. Area and volume, definition of a double integral on a more general set. Substitution of a double integral into polar coordinates, substitution theorem. Sequences and Series of Functions Sequences of functions, pointwise and uniform convergence. Theorem on the continuity of the limit of a uniformly convergent sequence. Power series, radius and disk of convergence. Term-by-term differentiation and integration of power series. Application to the summation of series.
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| Learning activities and teaching methods |
Monological explanation (lecture, presentation,briefing)
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| Learning outcomes |
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Elementary concepts of metric spaces, compact and complete spaces. Differential calculus of a real fuction of several real variables.
Functions of several real variables. |
| Prerequisites |
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Analytic thinking. AN2E.
KMA/KAN2 ----- or ----- KMA/PAN2 |
| Assessment methods and criteria |
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Oral exam, Written exam
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| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester |
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