Course: Mathematical analysis IV

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Course title Mathematical analysis IV
Course code UMB/572
Organizational form of instruction Lecture + Lesson
Level of course Bachelor
Year of study not specified
Frequency of the course In each academic year, in the summer semester.
Semester Summer
Number of ECTS credits 5
Language of instruction Czech
Status of course Compulsory, Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Berec Luděk, doc. Ing. Dr.
Course content
1. Vector functions of multiple variables: definition, basic properties, continuity, limit, differentiability, integration, examples (curves, surfaces, vector fields) 2. Vector functions of multiple variables: gradient, divergence, curl 3. Curves: the notion of curve, parametric equations of curve in plane and space, basic properties of curves (smooth curve, simple curve, closed curve, piecewise smooth curves), length of curve 4. Curve integral of the 1st kind: motivation for derivation, definition, derivation of calculation formula, examples, applications 5. Curve integral of the 2nd kind: motivation for derivation, definition, derivation of calculation formula, examples, applications 6. Green theorem, conservative vector field, independence of curve integral of the 2nd kind on integration path 7. Surfaces: the notion of surface, parametric equations of surfaces in space, smooth surfaces, piecewise smooth surfaces, tangent plane to parameterized smooth surface, area of surface 8. Surface integral of the 1st kind: motivation for derivation, definition, derivation of calculation formula, examples, applications 9. Surface integral of the 2nd kind: motivation for derivation, definition, derivation of calculation formula, examples, applications 10. Orientability and orientation of parameterized smooth surface, Gauss theorem (divergence theorem) 11. Stoke's theorem 12. Application of vector calculus: derivation of general conservation laws and of the Maxwell's equations of electromagnetic field in the form of partial differential equations

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Demonstration, Projection, Graphic and art activities
  • Preparation for classes - 42 hours per semester
  • Class attendance - 42 hours per semester
  • Preparation for exam - 42 hours per semester
Learning outcomes
To acquaint students with vector-valued functions, curve and surface integrals.
Student should have a clear idea of what vector functions are and be able to identify such functions in practical situations. Student should also know how to work with such functions, which means to have an idea about limits, continuity and differentiability of such functions and how to calculate their derivatives and integrals. Student should also be able to work with three types of vector functions, lines, surfaces and vector fields, including understanding and calculation of line and surface integrals. At the end of the course, student should be aware of where the obtained knowledge can be practically applied. Specifically students of physics should be able to connect the here obtained knowledge with knowledge obtained in courses of theoretical physics.
Prerequisites
Students are assumed to have knowledge of calculus covered by the courses Matematická analýza I (UMB564), Matematická analýza II (UMB565) and Matematická analýza III (UMB566).
UMB/CV566
----- or -----
UMB/566
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UMB/566K

Assessment methods and criteria
Combined exam, Test, Interim evaluation

80% attendance on practicals
Recommended literature
  • Colley SJ. Vector calculus. Pearson, 2012. ISBN 978-0-321-78065-2.
  • Colley SJ. Vector calculus. Pearson, 2012. ISBN 978-0-321-78065-2.
  • Jarník V. Diferenciální počet II, Integrální počet II. Academia Praha, 1984.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Physics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Applied Informatics (1) Category: Informatics courses - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Biophysics (1) Category: Physics courses - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mechatronics (1) Category: Special and interdisciplinary fields 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Physics (1) Category: Physics courses - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Applied Mathematics (2010) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer