Course: Calculus III

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Course title Calculus III
Course code UMB/566
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 winter semester.
Semester Winter
Number of ECTS credits 8
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Eisner Jan, Mgr. Dr.
  • Berec Luděk, doc. Ing. Dr.
Course content
1. Functions of several variables: practical examples, formal definition, sets in R^n, point neighbourhood in R^n, point classification in plane, domain and its determination, graph. 2. Limit: analogy and differences compared with single variable functions, independence on path, formal definition of limit, limit theorems, exmaples of calculating limits and proving their non-existence. 3. Continuity: definition of continuity at a point and on a set, examples. 4. Partial and directional derivatives: motivation, definition, computation rules, relationship between existence of a derivative and continuity at a point, derivatives of higher orders, mixed derivatives. 5. Differentiability: motivation and ideas leading to definition of differentiability of a function of several variables, differentiability and continuity, tangent plane, differential. 6. Derivative of compound functions: computation rules, directional derivative, gradient. 7. Implicit functions and their derivatives. 8. Taylor polynomial, local extrema of functions of several variables, necessary and sufficient conditions. 9. Absolute extrema, constrained extrema. 10. Double integral: definition, basic properties, Fubini theorem, basic transformations (translation, polar coordinates), method of substitution, geometric applications of double integrals. 11. Triple integral: definition, basic properties, Fubini theorem, basic transformations (spherical and cylindrical coordinates), method of substitution, geometric applications of triple integrals.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Demonstration, Projection, Graphic and art activities
  • Class attendance - 70 hours per semester
  • Preparation for classes - 28 hours per semester
  • Preparation for credit - 28 hours per semester
  • Preparation for exam - 56 hours per semester
Learning outcomes
Introduction to calculus of several variables.
Student should have a clear idea of what are functions of two or more variables, be able to identify such functions in practical situations, and be able to visualize graphs of functions of two variables. 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 partial derivatives and integrals. When working with functions of two variables, emphasis is put on geometric imagination and logical relationship of the presented concepts. In short, student should know what are functions of two or more variables and why is their study practically important.
Prerequisites
Students are assumed to have knowledge of calculus covered by the courses Matematická analýza I (UMB564) and Matematická analýza II (UMB565).
UMB/CV565
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UMB/565
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UMB/565K

Assessment methods and criteria
Combined exam, Test, Interim evaluation

100% attendance of practicals (excused absence permitted), performance of all written tests on practicals and at least 50% success in these tests
Recommended literature
  • Anton H, Vivens I. a Davis S. Calculus, 10th edition. John Wiley and Sons, Inc., 2012. ISBN 978-0-470-64772-1.
  • V. Jarník. 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): Applied Mathematics (2010) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Measuring and Computer Technology (1) Category: Electrical engineering, telecommunication and IT - Recommended year of study:-, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Winter
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: Winter
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: Winter
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: Winter
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: Winter
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Biophysics (1) Category: Physics courses - Recommended year of study:-, Recommended semester: Winter
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: Winter
Faculty: Faculty of Science Study plan (Version): Biophysics (1) Category: Physics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Measuring and Computer Technology (1) Category: Electrical engineering, telecommunication and IT - Recommended year of study:-, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Physics (1) Category: Physics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 2 Recommended year of study:2, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Mechatronics (1) Category: Special and interdisciplinary fields 2 Recommended year of study:2, Recommended semester: Winter