IMPORTANCE OF 3D ANIMATIONS IN MATHEMATICS II

Trendy ve vzdělávání 2012
Informační a komunikační technologie a didaktika ICT
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
Trendy ve vzdělávání 2012
Informační a komunikační technologie a didaktika ICT
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:
•
•
•
•
•
1
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
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
Trendy ve vzdělávání 2012
Informační a komunikační technologie a didaktika ICT
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
Trendy ve vzdělávání 2012
Informační a komunikační technologie a didaktika ICT
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
Trendy ve vzdělávání 2012
Informační a komunikační technologie a didaktika ICT
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: [email protected]
Lukáš Urbančok, student,
Obor Bezpečnostní technologie, Fakulta
aplikované informatiky UTB ve Zlíně,
Nad Stráněmi 4511, 760 05 Zlín, ČR,
e-mail: [email protected]
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: [email protected]
422