Course objectives:
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The aim of this course is an introduction and active understanding of the concepts the advanced mathematical analysis such as: - function sequences and function series; - vector functions of one real variable; - real functions of more variables; - differential and integral calculus in Rn. This course is lectured in English, its subject is equivalent to KMA/MA2. Students should be familiar with basic notions of mathematical analysis to the extent of the course KMA/M1 or KMA/MA1 or KMA/MA1-A.
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Requirements on student
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Demonstrate knowledge of the definitions, fundamental theorems and their proofs concerning function sequences, function series, vector functions of one real variable and real functions of more variables. Use rigorous arguments in calculus and ability to apply them in solving problems on the topics in the syllabus.
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Content
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Week 1: Point-wise and uniform convergence of function sequences; Week 2: Function series; Week 3: Power series and their convergence; Fourier series; Week 4: Vector functions of one real variable and their properties; curves in Rn; Week 5: Subsets of Rn and their topological properties; Week 6: Functions of n variables, their limits and continuity; Week 7: Directional derivative, total differential, tangent manifolds; chain rule; Week 8: Solvability of functional equations and differentiation of implicit functions; Week 9: Fundamental notions of min/max theory in Rn; Week 10: Mapping from Rn to Rm, its continuity and differentiability; regular mappings and transformations of coordinate systems; Week 11: Double and triple integral, Fubini theorem, basic techniques; Week 12: Application of double and triple integrals in geometry and physics; Week 13: Integrals depending on parameters and their differentiation. Further information and the lecture notes can be found on the web page http://analyza.kma.zcu.cz.
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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78
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Presentation preparation (report) (1-10)
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10
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Preparation for an examination (30-60)
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60
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Preparation for comprehensive test (10-40)
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36
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Total
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184
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
There is no prerequisite for this course. Students should be familiar with basic notions of mathematical analysis to the extent of the course KMA/M1 or KMA/ MA1 or KMA/MA1-A. |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
By the end of the course, a successful student should be able to:
1. Use notions of advanced calculus in English; 2. Demonstrate knowledge of the definitions and fundamental theorems concerning function sequences, function series, vector functions of one real variable and real functions of more variables; 3. Deal with function sequences and function series; 4. Expend a function into a power of Fourier series; 5. Describe curves in Rn and work with them; 6. Determine properties of functions of more variables; 7. Compute directional and partial derivatives of functions of more variables; 8. Formulate basic min/max problems and solve them using differential calculus; 9. Evaluate double and triple integrals; 10.Deal with integrals depending on parameters; 11.Use developed theory in solving problems on physical systems.
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills demonstration during practicum |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture supplemented with a discussion |
Interactive lecture |
Task-based study method |
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