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, Compulsoryoptional 
Form of instruction  Facetoface 
Work placements  This is not an internship 
Recommended optional programme components  None 
Lecturer(s) 


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 nonexistence. 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

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/565  or  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 

Study plans that include the course 
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