| Course title | Mathematical analysis IV |
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| Course code | UMB/CV572 |
| Organizational form of instruction | Lecture + Lesson |
| Level of course | unspecified |
| 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 | unspecified |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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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
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| Learning activities and teaching methods |
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Demonstration, Projection, Graphic and art activities
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| Learning outcomes |
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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 |
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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 ----- or ----- UMB/566K |
| Assessment methods and criteria |
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Combined exam, Test, Interim evaluation
80% attendance on practicals |
| 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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