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Lecturer(s)
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Marek Jaroslav, Mgr. Ph.D.
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Zahrádka Jaromír, RNDr. Ph.D.
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Course content
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Lecture topics by week of the semester: 1. Number series. Series with non-negative terms. Alternating rows. Absolutely and non-absolutely convergent series. 2. Power series. Taylor and Maclaurin series. Applications in integral calculus. 3. Vectors, matrices and tensors. Properties of matrices. Matrix operations. Linear and quadratic forms. 4. Solving systems of linear equations. Determinant of a matrix. Gaussian elimination method. Cramer's rule. 5. Inverse matrix. Eigenvalues and vectors. Sylvester's criterion. 6. Limit and continuity of functions of several variables. Total differentials of higher orders. Taylor's theorem and its use. 7. Local and global extrema of a function of several variables. Bound extrema, substitution method, method of Lagrange multipliers, Jacobian method. 8. Derivative in a given direction, nabla operator, gradient of a scalar field, divergence and rotation of a vector field. 9. Advanced games of differential calculus. Mean value sentences. Implicit function and its derivative. 10. Advanced parts of the integral calculus of functions of one variable. Special substitution. 11. Ordinary differential equations. General and particular solutions. Cauchy problem. Separation of variables. Homogeneous and inhomogeneous linear differential equations of the 1st order. Method of variation of constants. 12. Advanced parts of the integral calculus of functions of several variables. Transformation of variables. Double and triple integrals. Curve integral. Application. The content of the exercises corresponds to the topics of the lectures.
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Learning activities and teaching methods
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Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Work with text (with textbook, with book), Demonstration, Skills training
- Contact teaching
- 16 hours per semester
- Home preparation for classes
- 64 hours per semester
- Preparation for a credit (assessment)
- 48 hours per semester
- Preparation for an exam
- 52 hours per semester
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Learning outcomes
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The course aims to acquaint students with advanced parts of mathematical analysis. Emphasis is placed especially on understanding the main ideas of mathematical methods and the ability of students to solve practical problems.
After completing the course, the student demonstrates knowledge of mathematical analysis, specifically number series, advanced parts of the differential and integral calculus of functions of one or more variables, differential equations. The student is able to use the studied mathematical methods.
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Prerequisites
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Knowledge of the differential and integral calculus of functions of one and more variables.
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Assessment methods and criteria
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Written examination, Home assignment evaluation, Didactic test
At least 80% active participation in seminars + credit test (according to the teacher's decision in the form of Moodle or written). A maximum of 3 absences can be replaced by elaboration of a semester work. For repeating students: Recurring student can choose either a 80% participation and recognition test from last year, or at least 50% attendance and a new credit test. The course is completed by written exam, at least 50% of success is required. Classification E 50% - 55 %, D 56 % - 62 %, C 63 % - 69 %, B 70 % - 76 %, A more than 77 %. An oral form of the exam is optional, upon a student's request.
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Recommended literature
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AYRES, Frank a MENDELSON, Elliott. Schaum's outline of calculus. 6th ed. Schaum's outline series. New York: McGraw-Hill, 2013. ISBN 978-0-07-179553-1.
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CABRNOCHOVÁ, Renáta a PRACHAŘ, Otakar. Průvodce předmětem matematika I. (1. část), Úlohy z logiky, teorie množin a ze základů matematické analýzy. Vyd. 4. Pardubice: Univerzita Pardubice, 2008. ISBN 978-80-7395-112-2.
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CABRNOCHOVÁ, Renáta a PRACHAŘ, Otakar. Průvodce předmětem matematika I. (2. část), Úlohy z diferencionálního a integrálního počtu funkcí jedné reálné proměnné. Vyd. 5. Pardubice: Univerzita Pardubice, 2008. ISBN 978-80-7395-113-9.
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KOLDA, Stanislav, Ludmila MACHAČOVÁ a Otakar PRACHAŘ. Cvičebnice z matematiky II. Vyd. 9. Pardubice: Univerzita Pardubice, 2007. ISBN 80-7194-932-9.
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MACHAČOVÁ, Ludmila. Matematika: základy diferenciálního a integrálního počtu. Vyd. 6. Pardubice: Univerzita Pardubice, 2010. ISBN 80-7395-312-9.
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MAREK, Jaroslav; PASTOR, Karel a POZDÍLKOVÁ, Alena. Vysokoškolská matematika: výklad, řešené příklady a cvičení. Pardubice: Univerzita Pardubice, 2021. ISBN 978-80-7560-373-9.
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