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Lecturer(s)
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Dofková Radka, doc. PhDr. Ph.D.
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Laitochová Jitka, doc. RNDr. CSc.
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Course content
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- Differential calculus of functions of two or more variables. It is focused at applications of partial derivatives to find maxima and minima. Topics: n-dimensional space, metric space, Euclidean space. Neighborhood of n-dimensional space. Function of several variables. Domain and range. Geometric meaning of the function z = f (x, y). Limit of a function of several variables. Improper limit. Continuity of functions of several variables. Composite functions of several variables. Theorem on the continuity of composite functions. Partial derivatives of functions of several variables. Geometrical meaning of partial derivative of a function f (x, y). Higher partial derivatives. Schwarz theorem. Differentiable function. Complete differential. Geometrical meaning of the complete differential df(x, y). Complete differentials of higher orders. Partial derivatives of composite functions. Higher derivatives of a composite function. Taylor and Maclaurin's formula. Maxima, Minima, and Saddle Points. Fermat's theorem Sufficient conditions for local extrema. Implicit functions and their derivatives. Theorems on the existence of a derivative of an implicit function expressed by the equation F (x, y) = 0 and the equation F (x, y, z) = 0 - Infinite sequences and infinite series of constants and functions. Basic theory of infinite series. Applications of power series. Topics: Infinite sequences of numbers. Infinite series of numbers - basic terms and concepts. Series with non-negative members. Absolute convergence. Sequences and series of functions. Power series and their applications.
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Learning activities and teaching methods
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Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook)
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Learning outcomes
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The object of study - Differential calculus of functions of two or more variables. Applications of partial derivatives in search of extremes. - Sequences, series of numbers and series of functions. The aim is to present the theory of infinite series. It shows the importance and use of power series.
Finding extremas of functions of several real variables. Understanding the generalization of concepts introduced in calculus for functions of one variable to functions of several variables. Introduction to the theory of infinite series, which is one of the foundations of mathematical analysis. An overview of the applications of the theory of infinite sequences and series and ability to use the
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Prerequisites
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In addition to high school mathematics is assumed knowledge of differential and integral calculus of functions of one variable (Calculus 1, 2).
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Assessment methods and criteria
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Mark, Oral exam, Written exam
Passing tests, elaboration of homeworks.
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Recommended literature
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Došlá, Z., Došlý, O. (2006). Diferenciální počet funkcí více proměnných.. Brno: Masarykova univerzita.
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Došlá, Z., Plch, R., Sojka, P. (1999). Matematická analýza s programem Maple. Diferenciální počet funkcí více proměnných. Brno: Masarykova univerzita.
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Hájek, J. (1993). Cvičení z matematické analýzy.Diferenciální počet funkcí více proměnných.. Brno: Masarykova univerzita.
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Hájek, J., Dula J. (1990). Cvičení z matematické analýzy.Nekonečné řady.. Brno: Masarykova univerzita.
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Jarník, V. (1976). Diferenciální počet II.. Praha: Academia.
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Škrášek, J. Tichý, Z. (1983). Základy aplikované matematiky II. Praha: SNTL.
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Zuzana Došlá, Vítězslav Novák. (2007). Nekonečné řady. Brno.
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