IMPORTANCE OF 3D ANIMATIONS IN MATHEMATICS II EDUCATION FOR STUDENTS OF SECURITY TECHNOLOGIES STUDY FIELD AT TBU IN ZLÍN FIALKA Miroslav URBANČOK Lukáš CHARVÁTOVÁ Hana, ČR Abstract Nowadays a large number of students, whose geometric imagination and knowledge are not at sufficient level, are coming to study at the universities with technical branches. One of the most important tasks in the teaching of mathematical subjects is therefore to use all the latest software tools to improve this situation in the teaching process. In the article we deal with the modelling of 3D graphics using computer algebraic system Wolfram Mathematica in extremal problems determining the maximum or minimum values of functions of two variables. In general, the extremal problems are considered to one of the most important and most often solved problems in the theory and engineering applications. Graphs of functions of two variables can be already very complicated, and therefore geometrically very indistinct. That is why we demonstrate the extremal problems in the teaching of Mathematics II. We chose two types of surfaces, which can occur, at first the local maximum and at second the saddle point. The first case is a generalization of local maximum and the second of the inflection point that students already know from teaching at secondary school. Key words: differential calculus of several variables, Czech technical norm, animations, software Wolfram Mathematica. DŮLEŽITOST 3D ANIMACÍ VE VÝUCE MATEMATIKY II PRO STUDENTY OBORU BEZPEČNOSTNÍ TECHNOLOGIE NA UTB VE ZLÍNĚ Resumé V současné době přichází na vysoké školy technického zaměření velký počet studentů, u nichž není geometrická představivost i znalosti na dostatečné úrovni. Proto mezi nejdůležitější úkoly ve výuce matematických předmětů patří využít rovněž všech nejnovějších softwarových prostředků, jak tento stav ve vyučovacím procesu zlepšit. V článku se zabýváme modelováním 3D grafiky pomocí systému počítačové algebry Wolfram Mathematica v extremální úloze o zjišťování maximálních nebo minimálních hodnot funkcí dvou proměnných. Problematika extremálních úloh je všeobecně považována za jednu z nejdůležitějších i nejčastěji řešených úloh v teorii i inženýrských aplikacích. Grafy funkcí už i dvou proměnných však mohou být velmi komplikované, a tedy geometricky velmi nepřehledné. Proto jsme pro demonstraci extremální úlohy ve výuce Matematiky II zvolili dva typy ploch, kdy u prvního nastává lokální maximum a u druhého se objeví sedlový bod. První případ je zobecněním lokálního maxima a druhý inflexního bodu, který již studenti znají z výuky na střední škole. Klíčová slova: diferenciální počet funkcí více proměnných, Česká technická norma, animace, software Wolfram Mathematica. 418
Introduction Content of the Mathematics II subject at the Faculty of Applied Informatics of the Tomas Bata University in Zlín, which we present in the article, is often taught at other universities in the second term. The same situation is at our faculty, where this subject is supported by two textbooks. (Fialka, 2008a; 2008b) The both textbooks contain mathematic signs and symbols which are in accordance with the valid Czech Technical Norm. (ČSN ISO 31-11, 1999) Note only that the standard is often not even respected by authors of secondary school textbooks. Mathematics II includes the differential calculus of functions of several variables with applications. Among others this subject contains the following sections and topics: Notes to affine spaces and vector algebra Notes to metric spaces Sets of points, first of all, in Euclidean spaces Introduction to differential calculus of several variables Differential calculus of several variables 1 Wolfram Mathematica in the teaching of mathematical subjects When we solve various mathematic problems or for example at modelling of the graphs of functions, we use the computer algebraic system Wolfram Mathematica at the Faculty of Applied Informatics of the Tomas Bata University in Zlín. Each of students, who attends one of the three from seven faculties of our university, obtains the licence for this software system. Mathematica is widely well-known program for technical and scientific calculations. It is one of the environments with implemented tools for numeric and symbolic mathematics. The program fully supports the drawing of three-dimensional graphics and documentation in a notebook. Mathematica notebook allows us to create the complete technical documentation, which consists of formatted text, sounds, images, animations, hyperlinks, mathematic expressions, graphics, etc. 2 Samples of the 3D animations from the multivariable differential calculus The main benefit of presented three-dimensional animations of our contribution is its efficient use in the teaching of several variables calculus to the graphic clarification of the concept of partial derivative, tangent plane, gradient, local extremum and saddle point of a function of two variables at a certain point. Figure 1: The first two phases of animation of tangent plane to the Gaussian hat 419
The previous figure on the left shows the tangent plane to the Gaussian surface. This surface plays the key role in the probability theory, mathematic statistics and applications. The tangent plane is parallel to the coordinate plane Oxy. Mathematically, this means that the gradient of the Gaussian function is the zero vector at the considered point, whereas that point is the orthogonal projection of the top of the hat. This is an analogy with the differential calculus of one variable, where the tangent to the graph of a function at a given point is parallel to the axis, which maps an independent variable. Each presented animation is represented by means of two figures which show two phases of corresponding animation in Mathematica environment. The figures show the location and shape of the tangent plane to the graph of the function at the point, whose coordinates can be changed. Figure 2: The second two phases of animation of tangent plane to the Gaussian hat Depending on the position of a point, the tangent plane continuously skids in the both animations. At the same time the values of the angular coefficients of the appropriate tangents are calculated at a specific point. The angular coefficients represent values of the partial derivatives with respect to the variable x and y, respectively. In case of animations of the Gaussian hat, saddle point and also in other cases of surfaces time-consuming calculations appear. This causes a significant slowdown of the displayed animations. For this reason, the software offers the possibility to reduce the quality of a displayed function, thereby the response speed of the program can be significantly increased. Figure 3: The first two phases of animation of tangent plane to the hyperbolic paraboloid 420
Figure 4: The second two phases of animation of tangent plane to the hyperbolic paraboloid The authors used a modification of the graphical output in this article, which was created as a part of the bachelor thesis manual. (Talaš, 2011) The head of this work was M. Fialka. Authors of this paper would like to thank the author of the above mentioned manual for this material. Conclusion Teachers of the universities know very well that the use of the latest technology and software tools in teaching is essential. It positively affects students' access to the studied subject matter. Extremal problem, which we discussed in this article, representatively shows the connection between extremal problem of single variable calculus and multivariable calculus. Co-author L. Urbančok as a second-year student has a personal experience with animation presented in that teaching, which he attended last year. He personally confirms that for him as well as for other students of the same study field both presented animations were convincing and informative. These showed very clearly that the parallelism of the tangent plane to the graph of surface to the coordinate plane Oxy is only the necessary condition for the existence of local extremum of the function of two variables. Here, the saddle point is a generalization of the inflection point, which is known from teaching of mathematics at secondary schools. It is a well-known analogy of the differential calculus of one variable. In addition, the Gaussian surface of the normal distribution is also of considerable importance in the security technology. Gaussian curve or Gaussian surface of the normal distribution is used for example in the case of typological approaches to identify the crime offender. Bibliography 1. ČSN ISO 31-11. Veličiny a jednotky - část 11: Matematické znaky a značky používané ve fyzikálních vědách a v technice. Praha: Český normalizační institut, 1999, 27 s. 2. FIALKA, Miloslav. Diferenciální počet funkcí více proměnných s aplikacemi. 3. vyd. Zlín: Univerzita Tomáše Bati ve Zlíně, 2008a. 145 s. ISBN 978-80-7318-665-4. 421
3. FIALKA, Miloslav. Integrální počet funkcí více proměnných s aplikacemi. 3. vyd. Zlín: Univerzita Tomáše Bati ve Zlíně, 2008b. 103 s. ISBN 978-80-7318-668-5. 4. TALAŠ, Stanislav. Inovace výuky předmětu Matematika II na FAI UTB ve Zlíně elektronickou podporou obsahující ukázky řešení v prostředí Mathematica. Příručka k bakalářské práci. Zlín: Univerzita Tomáše Bati ve Zlíně, 2011. Assessed by: Ing. Bc. Bronislav Chramcov, Ph.D. Contact address: Miloslav Fialka, RNDr. CSc., Ústav matematiky, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, tel. 00420 576 035 002, e-mail: fialka@fai.utb.cz Lukáš Urbančok, student, Obor Bezpečnostní technologie, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, e-mail: l_urbancok@fai.utb.cz Hana Charvátová, Ing. Ph.D., Ústav automatizace a řídicí techniky, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, tel. 00420 576 035 274, e-mail: charvatova@fai.utb.cz 422