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Lecturer(s)
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Řezníčková Jana, Mgr. Ph.D.
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Course content
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1. Metric Space. Metric. Convergence of a Sequence in a Metric Space. 2. Open Set and Closed Set. Complete metric Space. Banach Fixed-Point Theorem. 3. Function of Several Variables. Graph of a Function of Several Variables. Neighbourhood of a Point. Limit and Continuity of a Function of Several Variables. 4. Partial Derivative of a Function of Several Variables. Directional Derivative. Gradient of a Function. Total Differential. Tangent Plane and Normal Line to Surface. 5. Higher Order Partial Derivatives. Higher Order Differentials. Taylor Polynomial. 6. Local, Global and Constraint Extrema of a Function of Several Variables. 7. Implicitly Defined Function of Several Variables. Implicit Differentiation. 8. Basic Properties of Double Integral. Evaluating Double Integral. Fubini's Theorem. 9. Transformation of Double Integral. Double Integral in Polar Coordinates. 10. Some Applications of Double Integral - Volume of a Solid, Area of a Region, Moment of Inertia of a Region in a Plane, Center of Mass of a Region in a Plane. 11. Numerical Methods for Solving Linear Algebraic Equations and their Systems. Iteration Methods. 12. Numerical Methods for Solving Nonlinear Algebraic Equations and their Systems. Simple Fixed-Point Iteration. Newton Method. 13. Interpolation and Approximation of Functions. Numerical Differentiation and Integration. 14. Solving First Order Ordinary Differential Equations and their Systems Using Picard method.
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Learning activities and teaching methods
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Lecturing, Methods for working with texts (Textbook, book), Demonstration, Projection (static, dynamic), Practice exercises
- Participation in classes
- 56 hours per semester
- Home preparation for classes
- 19 hours per semester
- Preparation for course credit
- 20 hours per semester
- Preparation for examination
- 40 hours per semester
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| prerequisite |
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| Knowledge |
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| Basic knowledge of linear algebra and differential and integral calculus of function of one variable is required. |
| Basic knowledge of linear algebra and differential and integral calculus of function of one variable is required. |
| learning outcomes |
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| explain the concept of a metric |
| explain the concept of a metric |
| explain concepts: a real function of two real independent variables, a domain and a range of a real function of two independent variables |
| explain concepts: a real function of two real independent variables, a domain and a range of a real function of two independent variables |
| describe the geometrical meaning of partial derivatives of a function of two variables at a given point |
| describe the geometrical meaning of partial derivatives of a function of two variables at a given point |
| explain the meaning of a gradient of a function of two variables at a given point |
| explain the meaning of a gradient of a function of two variables at a given point |
| give a procedure of finding local extrema of a function of two variables |
| give a procedure of finding local extrema of a function of two variables |
| present some applications of double integrals |
| present some applications of double integrals |
| explain the transformation of a double integral into polar coordinates |
| explain the transformation of a double integral into polar coordinates |
| Skills |
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| find and sketch a domain of a function of two variables |
| find and sketch a domain of a function of two variables |
| calculate partial derivatives of a function of two variables |
| calculate partial derivatives of a function of two variables |
| state an equation of a tangent plane of a function of two variables in a given point |
| state an equation of a tangent plane of a function of two variables in a given point |
| find critical points of a function of two variables and investigates local extrema of the function of two variables |
| find critical points of a function of two variables and investigates local extrema of the function of two variables |
| find absolute extrema of a function of two variables on a compact set |
| find absolute extrema of a function of two variables on a compact set |
| describe simple regions of integration (square, triangle, rectangle, region between two graphs) using inequalities |
| describe simple regions of integration (square, triangle, rectangle, region between two graphs) using inequalities |
| calculate a double integral in cartesian coordinates |
| calculate a double integral in cartesian coordinates |
| transform a double integral into polar coordinates and calculate it |
| transform a double integral into polar coordinates and calculate it |
| solve a nonlinear algebraic equation using a numerical method |
| solve a nonlinear algebraic equation using a numerical method |
| teaching methods |
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| Knowledge |
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| Practice exercises |
| Practice exercises |
| Projection (static, dynamic) |
| Projection (static, dynamic) |
| Methods for working with texts (Textbook, book) |
| Methods for working with texts (Textbook, book) |
| Demonstration |
| Demonstration |
| Lecturing |
| Lecturing |
| assessment methods |
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| Written examination |
| Written examination |
| Grade (Using a grade system) |
| Grade (Using a grade system) |
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Recommended literature
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ČERMÁK, Libor a Rudolf HLAVIČKA. Numerické metody. Vydání třetí. Akademické nakladatelství CERM, 2016. ISBN 978-80-214-5437-8.
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DAHLQUIST, Germund a Ake BJÖRCK. Numerical methods. Mineola, N.Y.: Dover Publications, 2003. ISBN 0486428079.
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DOŠLÁ, Zuzana a Ondřej DOŠLÝ. Metrické prostory: teorie a příklady, 3. přeprac. vyd.. Brno: Masarykova univerzita, 2006. ISBN 80-210-4160-9.
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KALAS, Josef a Jaromír KUBEN. Integrální počet funkcí více proměnných. Brno: Masarykova univerzita, 2009. ISBN 978-80-210-4975-8.
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KUBÍČEK, Milan, DUBCOVÁ, Miroslava a Drahoslava JANOVSKÁ. Numerické metody a algoritmy. Praha: VŠCHT, 2005. ISBN 80-708-0558-7.
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OSTRAVSKÝ, Jan. Diferenciální počet funkce více proměnných. Nekonečné číselné řady. Zlín: UTB, 2004. ISBN 80-7318-203-8.
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WEIR, Maurice D., Joel HAAS, George B. THOMAS a Ross L. FINNEY. Thomas' calculus, 11th ed., media upgrade. Boston: Pearson Addison Wesley, 2008. ISBN 978-0-321-48987-6.
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