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The second semester of the four-semester course on Applied Mathematics. Basics of linear algebra and matrix calculus. Differential and integral calculus of functions of several variables. Ordinary differential equations.
Last update: Houfek Karel, doc. RNDr., Ph.D. (02.05.2023)
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The course credit is awarded at practicals after passing three brief (60 min.) tests. The first test will cover Topic 1) of the Syllabus, the second test will cover Topic 2), and the third test will include problems from Topics 3) and 4). The dates of the tests will be announced during the practicals. To pass a test, a student must obtain at least 50% of the total points available for that test.
After getting the course credit at practicals, students can attend the final exams. These exams consist of written and oral parts, and they take place during the examination period. The written part (60 min.) comprises solving 2 practical examples from topics 1)-4). The oral part (60 min.) is a discussion of theoretical concepts (definitions and theorems from lectures) related to the examples in the written part. Last update: Ryabov Artem, RNDr., Ph.D. (18.02.2026)
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https://dl1.cuni.cz/course/view.php?id=16205 Here, after logging in with your SIS/CAS credentials, you can read/download supporting study materials: PDF documents with lecture notes and examples from practicals. The documents will be regularly updated during the semester. Last update: Ryabov Artem, RNDr., Ph.D. (28.02.2024)
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The requirements for the exam correspond to the course syllabus to the extent that was given in the lectures and exercises. Last update: Houfek Karel, doc. RNDr., Ph.D. (02.05.2023)
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The course covers four topics:
1) Linear algebra: Linear vector spaces. Matrices and determinants, systems of linear equations, Gaussian elimination. Bilinear and quadratic forms, positive and negative definiteness.
2) Basic theory of functions of several variables: metric, limits, continuity. Partial derivatives and total differential, operators grad, div, rot. Multidimensional integral. Exchange of limits and integrals, derivatives and integrals.
3) Series: Number series, convergence and divergence, absolute and non-absolute convergence, Taylor series.
4) Ordinary differential equations and their systems: basic methods, Bernoulli and Euler equations, equations in the form of total differential, solving equations using series. Last update: Ryabov Artem, RNDr., Ph.D. (28.02.2024)
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Upon successful completion of the course, the student will be able to:
1) Linear Algebra
Explain the fundamental concepts of linear vector spaces and their properties.
Work with matrices and determinants and solve systems of linear equations using Gaussian elimination.
Analyze bilinear and quadratic forms and determine their positive or negative definiteness.
2) Functions of Several Variables
Understand and apply the basic theory of functions of several variables, including metric spaces, limits, and continuity.
Compute partial derivatives and total differentials and use differential operators such as gradient, divergence, and rotation.
Evaluate multidimensional integrals and justify the exchange of limits and integrals, as well as derivatives and integrals, under appropriate conditions.
3) Series
Analyze numerical series with respect to convergence and divergence.
Distinguish between absolute and non-absolute convergence.
Construct and apply Taylor series for functions.
4) Ordinary Differential Equations
Solve ordinary differential equations and systems of ordinary differential equations using standard analytical methods.
Apply special methods for Bernoulli and Euler equations and equations in the form of a total differential.
Use power series methods to obtain solutions of ordinary differential equations. Last update: Ryabov Artem, RNDr., Ph.D. (09.01.2026)
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