Course: Analysis of Functions of Several Variables

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Course title Analysis of Functions of Several Variables
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)
  • Šimůnková Martina, RNDr. Ph.D.
Course content
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.

Learning activities and teaching methods
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
Learning outcomes
Metric spaces, functions of several variables.
Metric spaces, functions of several variables, series in the Gauss plane C.
Prerequisites
Mathematical Analysis 1, Mathematical analysis 2, Algebra and geometry 1, Algebra and geometry 2
KMA/AN1M and KMA/AN2M

Assessment methods and criteria
Oral exam, Written exam

Credit: Active participation on seminars + tests. Exam: writtten and oral
Recommended literature
  • Brabec, J., Hrůza, B.:. Matematická analýza II. Praha, SNTL 1986..
  • Černý, I:. Matematická analýza, 2. část. Liberec, TUL 1996..
  • Černý, I:. Matematická analýza, 3. část. Liberec, TUL 1996..
  • Dont, M. - Opic, B.:. Matematická analýza III - úlohy. Praha, ČVUT 1982..
  • Jarník, V.:. Diferenciální počet I. Praha, 1963.
  • Jirásek, F. - Čipera, S. - Vacek, M.:. Sbírka řešených příkladů z matematiky II. Praha, SNTL 1989..
  • Nekvinda M.:. Matematika II. Liberec, TUL 2000..
  • Sikorski, R.:. Diferenciální a integrální počet, Praha, Academia 1973..
  • Veselý, J.:. Matematická analýza pro učitele, I, II. Matfyzpress, Praha, 1997..


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester