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


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

Learning outcomes 
To acquaint students with vectorvalued 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  or  UMB/566K 
Assessment methods and criteria 
Combined exam, Test, Interim evaluation
80% attendance on practicals 
Recommended literature 

Study plans that include the course 
Faculty  Study plan (Version)  Category of Branch/Specialization  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 