| Course title | Analysis of Functions of Several Variables |
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| Course code | KMA/AFVP |
| Organizational form of instruction | Lecture + Lesson |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Winter |
| Number of ECTS credits | 8 |
| 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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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 |
Monological explanation (lecture, presentation,briefing)
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| Learning outcomes |
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Metric spaces, functions of several variables.
Metric spaces, functions of several variables, series in the Gauss plane C. |
| Prerequisites |
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Mathematical Analysis 1, Mathematical analysis 2, Algebra and geometry 1, Algebra and geometry 2
KMA/AN1M and KMA/AN2M |
| Assessment methods and criteria |
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Oral exam, Written exam
Credit: Active participation on seminars + tests. Exam: writtten and oral |
| Recommended literature |
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| Study plans that include the course |