Symbolic computations in modern education of applied sciences and ...

Computer Assisted Me hani s and Engineering S ien es, 15: 143163, 2008.

2008 by Institute of Fundamental Te hnologi al Resear h, Polish A ademy of S ien es Copyright

Symbolic computations in modern education of applied sciences and engineering

Mar in Kami«ski

Chair of Me hani s of Materials, Te hni al University of Šód¹ Al. Polite hniki 6, 90-924 Šód¹, Poland

(Re eived August 18, 2008) A presentation of the modern issues related to the symboli omputing is ontained here together with the detailed dis ussion of its appli ation to the edu ation of various s ienti and engineering a ademi dis iplines. The future expansion of the symboli environment is des ribed here on the basis of their histori al and modern developments presentation. As it is shown on the example of the MAPLE system, symboli omputational environments play the very important role in supporting the le tures and the

lasses in the omputer labs. Those environments may be also very useful in tea hing basi natural s ien es in all those ases, when some algebrai or dierential equations appear, must be solved and their results should be pre isely dis ussed. The appli ation of the MAPLE and similar omputer systems in the engineering edu ation seems to be unquestionable now and some examples are ontained here to show how to improve the le tures and make them very interesting and ex iting. The key feature oered by the symboli omputing is the opportunity to dis over the knowledge that the students may do by themselves, when they are spe i ally leaded by the instru tors. Keywords:

symboli omputations, omputer s ien e, omputers in edu ation

1. GENERAL INTRODUCTION

Nowadays, a role of the omputer-based learning in all s ienti and engineering dis iplines and subje ts still seems to be de isive and is not limited of ourse to the symboli omputing programs appli ation. Those omputer methods are of ourse very widely understood and begin with the internet browsing and sear hing for some spe i knowledge sour es (the advan ed sear h allows to train logi operators). The tea hers and le turers should fo us the students not only on nding the answer for some spe i problem but, rst of all, on the reliability of the parti ular internet sour e. Considering a ommon pra ti e of nding everything, whi h is needed to prepare the homeworks at the s hools and universities on the relevant webpages, the tea hers inuen e is pra ti ally strongly limited to a veri ation of those sour es (without the opportunity of the ee tive orre tions of the net sour e in its ele troni version). The se ond pro ess stri tly onne ted to the above is an explosion of the omputer software (well pa ked into the graphi al environments). Starting from 3D games through business pa kages till the s ienti advan ed programs we an all use, modify or even build any ne essary pro esses ows to

omplete most of our work to be done on (and mostly) by the omputers. Therefore, a role of the modern tea hing in this ontext may be (and really is) limited to a reation of the written or spoken manuals to the spe i omputer software. In this ontext the Latin idea that `repetitorum mater studiorum est' transforms to the repeated usage of the same program or just the algorithm. Anyway, it may result with no doubt in a omplete disability of our hildren (and us too) to remember orre t grammar stru tures, for instan e. So that, the omplete omputer preparation of the homeworks for the English or history lessons should be stri tly prohibited. No one at this moment may prognose the so ial progress under su h ir umstan es (the written language may be totally left for the omputers as well in the nearest future).

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The edu ational aspe ts of the ` omputerization everywhere' have nevertheless many positive aspe ts, espe ially for the basi natural and engineering s ien es. The basi and the very lear dieren e to the previous bla kboard methods is the opportunity to (1) present the solution of the parti ular problem in its nal form with its overall veri ation, (2) ompare on the same s reen the solutions of the similar or even dierent problems, (3) study parameter sensitivity of those solutions and, espe ially (4) make a dynami visualization of our problem solution. So that we, as the tea hers, may pa k the ompli ated (boring for the students) problems together with the ne essary knowledge as the attra tive olorful boxes, whi h the hildren and the students want to open with no delay. It was quite impossible before and must be exploited now as widely as possible. Thanks to su h an edu ational method a large portion of the dierential equations appears as a quite natural prolongation of the well-known elementary algebrai equations, where some visible and tou hable (on the s reen) fun tions give the nal answer. Most of the le turers of the omputer s ien e ourses have wrong feeling that their ourses, thanks to the presen e of the hardware, olorful software and powerful 3D graphi s and animations are so interesting that they need no spe ial attention or separate edu ational tri ks to fo us the students mind on this subje t. It is true for the several number of tea hing hours when the students do not have the ability to use the omputer as they want. When they be ome to be uent, they start to be bored by the traditional omputer s ien e ourses, they start to browse the internet in a quite opposite dire tion, sometimes even to play the games et . The arguments that the design pro ess or the omputer-based analysis are very important, must be introdu ed in a proper way, sometimes

annot be followed step by step and needs even more attention than the traditional al ulations whi h are frequently ignored by the students. An attra tive pa kaging of any problem is relatively easier here than for the traditional ourses, far from the omputer methods, but anyway need the a ademi tea hers attention. There is no doubt that the symboli software reated the brand new opportunities for the le turers, whi h are demonstrated and dis ussed below. Su h pa kages like MAPLE [3℄, for instan e, are present in the regular ourses of omputer s ien e at many universities, however thanks to the re ent progress in symboli software development the role of this software essentially hanged last years. It is ree ted by in lusion of the symboli software into the mathemati al handbooks and textbooks [1℄. One of the most re ent and most apparent edu ational appli ations is providing the le tures or even the entire ourses in symboli algebra programs, whi h an in lude numeri al veri ation of more important equations together with their straightforward derivation. Su h a ourse may be saved in this pa kage format, so that having this system available at all omputer labs, the

ourse an be distributed by email (a le ture by le ture) to all parti ipants to be used during the

lasses. Let us underline that most of the illustrations are prepared in the symboli environment of the program MAPLE, however, symboli systems are nowadays so similar that those illustrations

an be translated without any loss to any omputer system of this type. Considering above, we would like to address the following issues in this paper  do the symboli

omputer systems oer something qualitatively new, how to tea h e iently the usage of symboli tools, how to orrelate ee tively the symboli omputing with the edu ational needs, how we an modify symboli programs to arry out the resear h and the le tures as well as how to intera t with symboli environment software ompanies. The dis ussion starts in the next se tion with the presentation of general aspe ts of the omputer s ien e tea hing explaining the basi rules and observations dealing with the omputer s ien e a ademi ourses as well as a dis ussion on the role of internet and games in edu ation of the young people nowadays. The third se tion ontains some introdu tory omments and the several histori al fa ts about the symboli omputing together with short omments on the future development of the omputer algebra systems (CAS). Both next two se tions ontain the detailed des ription of the symboli programs examples that an be used for lower level ourses (fth se tion) and for the PhD students (sixth se tion); the edu ational

omments are added on the o

asion of ea h of those examples. The paper is nished with the

on luding remarks following the onsiderations drawn from this work.

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2. GENERAL ASPECTS OF COMPUTER SCIENCE TEACHING 2.1. Various methodologies of omputer s ien e tea hing

The omputer s ien e and all omputer-aided ourses (from typi al for all the engineering ourses to fundamental s ien es like applied mathemati s) should have the spe i edu ational methods ree ting the fa t that they are provided in a omputer lab. So that they are based on proje ts or exer ises solved by the students during the lasses and ompleted by themselves at home. The edu ational strategies are quite dierent, depending on the university rank, the expe tations of the le turer, the abilities of the students, the ourse type and the presumed knowledge level of those students. Usually, the following s hemes are applied at the omputer labs: 1. the students is taught by an observation of the le turer presentation given during the lasses; analogous problems are left for the homework; 2. the student follows on his/her omputer the tea her presentation given during the lasses step by step; 3. the le turer explains the entire problem (or some lass of the problems further distributed amongst all students) and give a solution algorithm during the presentation or, after a short break, on the bla kboard (bla kboard explanation seems to be similar to the same algorithm

reation by the student on the paper sheet); 4. the le turer explains the whole problem but the algorithm is left for the students' ideas and initiative - further programs and omputer a tivities during and outside a lab follows this riti al point; 5. the le turer omments the key points of the problem to be solved by students only and gives on the bla kboard all ne essary equations to solve it; 6. the le turer explains the key points of this problem and leaves nding of all equations to the students (using their knowledge from previous ourses, ourse books and/or internet sour es); 7. the le turer explains only the whole problem, where the solution, its algorithm, equations to be implemented to nd this solution (and possibly its physi al interpretation) are the students' job; 8. the le turer may be the exer ises distributor only and then, the students must provide the entire solution (but the problem is generally known from the previous or parallel ourses); 9. the student works alone in the omputer lab having the problem (or a list of problems to be solved) available in the net or given at the beginning on the paper sheet; 10. the students may work at home, giving the presentation temporarily or at the end of the entire

ourse during some kind of the nal graduation. All those methods an be used in dierent ombinations, where the nal de ision (to assure the best ee tiveness of the ourse) depends on the le turer, the students level, the di ulty level of the problems to be solved and on the intera tion between the students and the tea her. Firstly, let us note that the rst is a rather wrong way be ause the le tures be ome very fast a more or rather less interesting lm and the students sitting behind the omputers start to do something else. So that most of the knowledge summarized in the presentation must be repeated, whi h leads undoubtedly to the essential time waste. The se ond option is not so dangerous, but anyway the mental s heme repeated lass by lass is that the student an still ount on the le turer, and there is no essential need to do everything by himself; ee tively some portion of the exer ises is done by the le turer, whi h does not belong to this le turer expe tations rather. The third possibility

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should appear during the lasses  at the rst time  to show the students the ne essity of the good algorithm reation and its optimization  but only on e  to avoid the situation that the le turer will deliver the algorithms for all problems and to introdu e the students as soon as possible into the algorithm reation by their own. The situation that the tea her explains in detail the whole problem to be solved together with all ne essary equations may appear but only on e or twi e  the problems should deal with the knowledge given to the students before, during the pre eding a ademi or even high s hools ourses, so that there is no real need to bring them all equations ea h time and to reate a rule that some more important s ienti problems will be explained during the omputer s ien e lasses and, furthermore, that there is no onne tion between dierent s ien e bran hes. This is not the very re ommended way in the age on inter onne tions between even extremely far s ienti dis iplines, where omputed s ien e ould be some platform to make them loser. The next option, where the le turer omments the most important issues seems to be better approa h, but it is good to perform it several times only and not always, of ourse. The help to the students during the lasses seems to be good in most ases, espe ially to understand the riti al points do not belonging to the omputer s ien e itself, but, on the other hand, the tea her must assure the students that the omputer-aided solution of any problem must in lude deep understanding of this problem out of the omputer and is not a omputer game; therefore, this option seems to be ase-sensitive. Hen e, the next option, when the le turer explains only the problem and the rest of the solution is left to the students' work seems to be the most justied and the most frequently applied edu ational s heme. The next options (8, 9 and 10) are provided for better students only, who are believed to work really alone; espe ially the last option onsisting in an almost permanent typi al homework should be given for the best student only as an honor. Whatever and however the le turer will nally adopt as his approa h to the students, it is very useful to remember that the students (a) should be very well motivated (not only by the good grades), (b) must be appre iated (not on every o

asion but anyway), ( ) may be fo used on ea h problem (even the most boring) by its interesting unusual presentation, (d) they simply like to ompete. One an a

omplish those goals by treating traditional exer ises like the ompli ated unsolved resear h problems, so that the students believing that they make the brand new dis overies are strongly asso iated to the given exer ise. Of ourse, there is no optimal edu ational methodology independent from many reasons, both predi table and unpredi table, however, treating the students groups statisti ally, the most re ommended way seems to be a ontinuous transition from the 3rd point to the 8th or 9th point from a le ture to le ture. Then, we start the ourse from the extensive explanation of almost any a tivity and, by a systemati elimination of some a tivities performed or repeated uently by the students, we tend to the situation when everybody works (in luding a le turer) alone at the lab. It is a good moment to underline a very interesting role of the in ompletely dened problems we an use for the se ond part of ea h ourse, when the students have been taught basi methods and tri ks. The solution pro ess of su h problems may be dis reetly observed by the tea her sin e the students to omplete this pro ess need to ask for some parameter, assumptions or fa ts, so that it is possible to make the alendar who asks rst and when, to distinguish between better and worse students et . Finally, let us remind that there are several ways that annot be followed a

ording to the opinion of many experien ed tea hers during the omputer s ien e labs and le tures: 1. The problems should not be given as the pi tures  espe ially for the engineering ourses  we do not solve pi tures but the real problem, where the right s heme reation may be even the most important part of the solution pro ess. Instantaneous graphi al presentation of the problems leads to the omplete la k of understanding what is really solved. 2. When all the students will get ompletely dierent problems they will not intera t with ea h other; it prevents inter hanging of the amera-ready solutions between the students (to eliminate this phenomenon we an simply dismiss the mailing tools usage during the lasses) but, at the same time, it an els any intera tion of the students, whi h is important  explanation of the

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moment whi h some students annot solve or just some questions to the olleague at the left or right needs some understanding of what we really do now. Let us also remember that: 1. The nal result should preferably ontain the gures with titles and omments, at least the few senten es ommenting the results obtained and their agreement with the initial expe tations, the existing knowledge and the ommon sense; it would be very worthy to demand some parametri studies from the students  it gives a lot of the so- alled `engineering intuition', whi h extends their imagination of the given proje t. 2. The entire methodology and our expe tations for the problem solution will introdu e the students into the s heme, whi h next may be exploited during the real life problem solutions and an avoid any failures or mistakes during their engineering or resear h pra ti e. At the end of this se tion we an re ommend the following problem, whi h an be solved by the students of many engineering ourses using symboli omputer programs and during the omputer s ien e lasses (if, of ourse, the theoreti al me hani s ourse was taken before). Problem: Solve the following equation, give its physi al interpretation, make its additional visualization, perform the parametri studies with respe t to m, c and k , provide the available animation. What is the physi al meaning of the rst and the se ond time derivatives of the solution?

m

d2 x(t) dx(t) +c + kx(t) = f (t). 2 dt dt

Ea h student may get here dierent fun tion f (t) but everyone should: (1) identify this equation as the 2nd order ordinary dierential equation, (2) identify its physi al meaning as the vibration of the single-degree-of-freedom system, (3) dene properly the right hand side fun tion and to introdu e the initial onditions (rst he k point), (4) make a visualization and dis ussion of the results, (5) give a physi al meaning of the velo ity and a

eleration (the 2nd he k point) and nally, (6) produ e the animation. 2.2. The role of internet and games in edu ation

Although initially those two subje ts were not very losely related to ea h other, today most of the games are found in the internet themati websites and, rst of all, may have some bad inuen e during the a ademi edu ation. The ommer ialization and the publi a

ess to both phenomena of the virtual nature may have some unwanted, unpredi table and even bad inuen e on the students, so that they need some a ademi attention. As it was mentioned in the previous onsiderations, the internet knowledge sour es play very important role in the up-to-date omputer-based learning, sometimes even de isive but not always in a positive meaning. A

ording to our observations in the omputer labs and in the a ademi libraries, the ele troni sour es almost repla e written sour es, whi h were traditionally reviewed or at least veried at the printout stage. Now, when the students must solve some problem very fast having

ompletely no or very limited knowledge on some subje t, they employ some popular web browser by putting some phrases or their various ombinations; it an be for instan e www.wikipedia. om. Negle ting the web page address and their inventors and internet providers maintaining this knowledge sour e, its veri ation pro ess onsists in further solution of the initial problem using this knowledge and presentation of the results to the le turer. When the le turer approves this solution on e, sometimes twi e, this web page (or the address) be omes more and more reliable and they are used without any further authorized approvals. This phenomenon is so popular now that some parts of the students populations believe without any doubts in all internet sour es even more than in the a ademi le tures or textbooks. It is not sensible to point out how dangerous it is that the

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improper fa ts and improper sour es migrate between the whole groups of the students, whi h an be dete ted during the nal exams, when the students may frequently re all that some spe i part of the material has been found in the internet. Sometimes the students by themselves solve some problems supposed to be the subje t of the forth oming exams and put them into the new private (se ret) websites. It needs of ourse further intensive attention of the le turers and the a ademi authorities how to prevent this situation. Undoubtedly one may reate and re ommend simple themati edu ational websites, where the entire department may olle t all the le tures, presentations together with some solved examples to shorten internet browsing amongst the students and to eliminate the possible errors. It does not mean of ourse that the internet plays bad role in the edu ational pro ess, when the le turers annot reate the themati internet sour es, they are always able to re ognize the most wanted addresses by the students and to spend some time during the lasses to verify the fa ts ontained on those pages. This problem seems to be rather trivial but the a ademi pra ti e is that it is usually omitted during the edu ation of omputer s ien e almost at all. This is not a se ret that one of the most powerful engines of the omputer business development is the omputer game industry and that the young people are very sensitive today to those problems playing frequently dierent games  at home, at the o e, at the university and sometimes even during the lasses in the omputer labs. The power of this phenomenon is quite lear on the mobile phones market and small ele troni devi es designs like the very modern last days PSPs. It is needless to say that the games (outside mathemati al games theory) are not in luded in any omputer s ien e

ourse and, at the same time, that the students try to play during the omputer s ien e labs be ause of an easy a

ess to the reliable and su iently fast omputer equipment. Negle ting the fa t that su h an a tivity means the serious time waste and is in onvenient during a ademi a tivity time, there is some mu h more serious reason to ontrol the time spent behind the games and, rst of all, their hara ter. Postponing a typi al lassi ation of dierent games available in the net, the tea hers should distinguish between the games having neutral mental inuen e on the students and young people in general (like s hemati playing games just to have a break), positive inuen e on their mentality (logi and strategi games) and, the most important, very lear negative (sometimes statisti ally proven) mental impa t. The urgent attention is ne essary when the students unusually use to play the ghting or war games with the multiple lives option. Those games give, thanks to their very attra tive 3D options, the feeling that the player remains in some fantasti virtual reality where he is to kill the enemies having guaranteed the next life in the ase of some failure. Naturally, the students interested in su h games try to ross the next and next levels trying to be as e ient as possible, to be faster and faster and, nally, to ompete with ea h other. The more they play, the virtual reality be omes

loser to the reality and, during longer sessions, it starts to intera t and to mix with the real life (not only a ademi ). It is observed during the exams and the proje ts evaluation that they start to treat the edu ation pro ess like the next game with the next life option, where some failure at any level does not really matter, sin e there is still a next option. The next step is noti ed when the tea her is supposed to make a mistake when this tea her is not in a position to oer the next

han e to have an evaluation. Therefore, the real life appears to be a weaker version of this game sin e no option to orre t this mistake and, whi h is essentially more dangerous, that the errors

annot be simply an elled from the overall memory by a simple resetting of this bad situation. This player mentality with a permanent interferen e of the virtual and the real worlds is espe ially visible when the students having some errors in their proje ts bring on e more the same proje ts without any orre tions to the supervisor expe ting that he will not noti e them (taken from a personal experien e of the author). Then, the next life and the game intera ts with the so ial onta ts with the supervisor  most probably, if he noti es that he will have the next option (my next life from the game) to orre t them then everything (like after the spe ial button pressing) remains the same as before. As it is widely known, the key moment of the engineering studies is in the proje ts preparation, mainly using the omputers, where this type of a mental behavior leads to the disaster that any error may be orre ted and has no real inuen e on the nal result. The proje t is also

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a kind of a game, be ause it is arried out on the same virtual platform, so that any mistake is not really important sin e there is the next life. It is not ne essary to explain the engineers how dangerous it an be in real designing pro ess, where a single mistake of the designer an ause a total disaster and annot be avoided on a ertain manufa turing phase. It is hard to imagine a population of the people having a ademi edu ation and thinking in the way that all humans' a tivities are the subje t of a game with the total renew options always a tive. There is a single, but important, ex eption from this rule and these are the ght simulators in military s hools, where the students must be prepared to ght and to train the war games with the enemies like terrorists. Con luding those fa ts and observations, we must say that enormous interest in the games with multiple lives options is dangerous for the students, must be prohibited during the edu ation of

omputer s ien e. Furthermore, having in the mind that the omputer games are so popular and they are not bad as a rule, it is advised to spend the few minutes to re ommend the students the most thoughtful (and attra tive at the same time) omputer games and to explain the students a motivation to his hoi e. It may positively inuen e the le turer position. As the example may serve the game alled HEX (easy available as the freeware HEXY) invented by Ameri an Noble prize re ipient John Nash, whi h re eived the gold medal on the omputer Olympi Games in Athens in 2000. It has a big han e to be a popular ompetition be ause a history of John Nash be ame popularized last years by the separate inema produ tion. The game is somewhat similar to the histori al game GO, but the dimensions of the rhomboidal game plane an be modied by the player, who, having the stones with the spe i olor, needs to reate a path form one side of his

olor to the other (move by move with the omputer as the se ond player).

Fig. 1.

Hexy s reen

3. THE GENERAL NOTES ON SYMBOLIC SOFTWARE AND ITS EVOLUTION

A

ording to various denitions, the symboli omputing is a dis ipline fo used on the representation and a manipulation of the information in the form of a symbol, whi h is mainly onne ted with its numeri representation for further pro essing. It is programmed to automate ompli ated algebrai manipulation with su h equations, also dierential and integro-dierential equations. The symboli

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omputer programs oers a large variety of the mathemati al internal fun tions and pro edures together with graphi al options providing a programming language to dene some users pro edures. From the te hni al point of view, the symboli software is implemented as the dire ted, aperiodi graphs but this programming algorithm and the ode remain still unavailable for the users at all. A history of the symboli software development starts at 1960's, thanks to the pioneering work of the Nobel Prize re ipient Martin Veltman who programmed a prototype of this software known as S hoonship. The next, more professional systems were developed at the MIT at 1970's and early 1980's; the rst most popular systems were known as muMATH, Redu e, Ma syma. The half of eighties brought the ommer ialization of the symboli pa kages and the next, still popular even today implementations like Maple, Math ad, Mathemati a or Maxima. The migration of the symboli software to s ienti al ulators was also noti ed at this time like rstly in the ase of Hewlett-Pa kard with its HP-28 having arrangements of the algebrai expressions, dierentiation, limited symboli integration, Taylor series expansion and the entire solver for algebrai equations system. Later on, Texas Instruments programmed its al ulator TI-92 with the use of Derive pa kage. Nowadays, the symboli software is available in a plenty of realizations for dierent operating systems (Linux, all Windows versions, Ma OS, for instan e), for grid omputing (like gridMathemati a, v. 6.0, sin e 2007), in some national versions (see Polish Derive sin e 2005). A ommer ialization level and the market for those programs may be well illustrated by the fa t that Mathsoft programming the system Math ad has been a quired by Parametri Te hnology Corporation for 62,0 million USD in April of 2006. The progress of the symboli software is visible in the three next slides (rst two from the internet Math ad sour es), where the evolution from the DOS-based simple fun tions operation and 2D visualization for one of the rst implementations of Math ad (Fig. 2), we browse through one of its rst Windows-based releases in 1992 (Fig. 3) to the re ent advertisement of the system Maple (available from www.maplesoft. om), where the entire designing methodologies have been in luded into this program (Fig. 4). It is lear that the symboli environments evolved from the typi al programming ompiler towards the typi al Windows appli ations with the buttons, the s rolled down menu and the very useful help options  this evolution progressed in parallel to the other types of software but the primary idea remained the same.

Fig. 2.

One of the rst releases of Math ad

Symboli omputations in modern edu ation

Fig. 3.

Fig. 4.

The main window of Math ad in 1992

Maple apabilities for its latest version 12.0 available sin e 2008

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The future of symboli omputing is not always in personal and massive omputers but also, maybe mainly, in popular portable devi es implementations like mobile phones, modern s ienti

al ulators and PDAs. Let us note at least (1) the TVH-72g Graphing Cal ulator works for the Sony Eri sson J300i ell phone, (2) Mobile Voodoo  the PalmOS based emulator for the HewlettPa kard 48SX, 48GX and 49G series of al ulators, (3) qdCAS  a omputer algebra system for the po ket personal omputer, (4) X as ARM works for some PDA devi es with Linux and Windows. It is ompletely useless to dis uss whi h way would be the most preferred, sin e various hardware implementations of the CAS programs are needed for essentially dierent s ienti and engineering problems  from more advan ed al ulations and visualization for the students during lasses, through some hanges in the engineering proje ts made in situ (also by the portable devi es) and the elementary s ienti problems for the edu ation visualization (on the lassi al PCs) to the real large s ale s ienti problems and simulations to be arried out on the multipro essor massive

omputers. The symboli omputing be ame so large last years that there exists in the internet the so- alled Symÿoli Net maintained by the Kent State University (http://www.symboli net.org) ontaining the

onferen es and other events, open positions, software, downloads and sear h engine as well as the subs ription sub-site. The software available in the symboli omputations area obey a plenty of the ommer ial pa kages like MAPLE, MACSYMA, MATLAB, MATHCAD or MATHEMATICA as well as even freeware like the js l-meditor (the java symboli omputing library and mathemati al editor, http://js l-meditor.sour eforge.net), MAXIMA or SCILAB (http://www.s ilab.org). All the detailed onsiderations ontained below are referen ed to the system MAPLE (present version 12.0) implemented for the rst time in 1981 by the Symboli Computations Group at the University of Waterloo in Canada. Nevertheless, there is no loss of a generality in dis ussing everything on MAPLE and all the options re alled here do exist in the remaining systems and are easy available in the remaining systems after small semanti modi ations only. Let us remind now that the genesis of all those programs is more less the same  they have been initiated at the applied mathemati s and/or omputer s ien e departments of the worldwide leading universities and through those years went out outside a ademi area by ommer ialization or are

Fig. 5.

The maplet builder window

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still being developed by the s ientists or engineers in many ountries under the GPLs. Nowadays, the symboli programs are very powerful and, besides all simple algebrai transformations and differentiation or integration te hniques, have the entire ODEs and PDEs libraries, extended statisti al as well as fantasti visualization and animation options. Last but not least, it is possible to build a separate window-released programs in luding longer sequen es of symboli operations devoted to a solution of the spe i problem (the so- alled maplets  see Fig. 5). Now, sometimes ompli ated semanti s is repla ed with the olle tion of various buttons and s rolled down menus, whi h perfe tly ree ts the modern tenden y to provide a visual form for most of the ne essary omputer operations. The maplet appli ation builder may motivate the students very mu h, be ause it is very attra tive itself as the new separate window appli ation, so that its reator is a good programmer and, on the other hand, the entire designing pro edure may be hidden parametri ally under a single window so that the parameters values modi ations may result in automati re al ulation of the entire (even extremely ompli ated) pro edure and graphi al presentation. It gives the quite new opportunity for the students, resear hers and s ientists  it is not really ne essary to know the MAPLE programming language to use it e iently to solve some problems and to make the results visualization.

4. EDUCATIONAL EXAMPLES FOR THE UNDER AND POSTGRADUATES

Now, let us show the sele ted, very important for quite dierent reasons, MAPLE appli ations, whi h an be used during the mathemati s or theoreti al me hani s lasses. This short review starts from the integration methods illustrations by the additional maplet shown below. It is obvious that the mathemati s hidden under the i ons is indeed very dangerous edu ational dire tion sin e our students be ome totally omputer-dependent mathemati ians and without the aid of a ma hine they

an do nothing of ourse. A paradox of this phenomenon is that the symboli programs rst redu ed mathemati s to the ommands and then, the symboli programmers implemented the windows-based appli ations builders to reate the edu ational appli ations demonstrating various mathemati al methods and its analyti al tri ks. The maplet for the integration by parts is visible below  see Fig. 6; a pure appli ation of the symboli integration pro edure does not demand of ourse any knowledge about all the methods available. Then, the students taught this operation without any methodologi al ba kground le ture (just `int' option) may think (and they really do so) that this pro edure is so easy that this is the real `pie e of ake' and there is no need to deal with it longer than a minute. Let us note that the essential value of this method is that the integration an be repeated reliably by the students at home  a

ording to the time limitation it is impossible to repeat this pro ess even in the omputer lab. So that, training this pro ess at home, the students may dis over by themselves some well-known rules, resulting in some primarily primitive mathemati al intuition, whi h in this ase is as important as the fundamental knowledge. Contrary to the opinions of many mathemati ians, it is not possible at all to obtain the same edu ational results using the bla kboard only, be ause a limited number of the problems that an be solved using this methodology do not lead to the on lusions that there are quite learly re ognizable types of integrals, whi h, after some time spent behind the omputer appear to be trivial. Sin e we do this integration in a step by step fashion, we an dete t how dierent methods intera t during the integration of the produ t of some basi fun tions. The students may easily hange the integrated fun tions by themselves and train this integration pro ess like a game. Next, we have the example how to easily al ulate the surfa e (as well as their volume next) for the user-spe ied surfa e of the revolution, whi h is integrated with the extended graphi al options. All may understand rather easily from this window-based internal appli ation within the MAPLE system the main general on ept behind this al ulation (see Fig. 7). Contrary to the previous example, we train here the integration over the interval, so that we an modify this interval dire tly, we an make a visualization of this integral rotated over a spe i axis,

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M. Kami«ski

Fig. 6.

The result of an integration in MAPLE

we may modify the initial fun tion and the line of revolution as well as the distan e of a rotation line to the spe ied oordinate axis (horizontal or verti al). The integral value appears in this window shortly after the integral to be omputed on the basis of the user dened number of the partitions in the Riemann sum. So that, we an demonstrate that by in reasing the number of partitions we obtain the onvergen e to the spe i limit. Let us underline that the bla kboard methodology an be based on rewriting of the basi formula and the omputation of the single or two examples. It is not possible at all to make even similar gure on the bla kboard, whereas the entire explanation needs a separate lasses, so that having this into a single user-modied window we an expe t from the students more understanding of this parti ular problem than before. The next appli ation of the MAPLE  to the vibrating me hani al system is shown here, where the solution is rather easy and the well-known (Fig. 8, with the amplitudes given parametri ally in Fig. 9). The parameters of this system dened as follows: m[M℄=1000 kg, m[U℄=500 kg, d=100 000 N/m, whereas the basi solution of the dierential equation itself redu es to the following three lines only:

restart; with(DEtools); sol := dsolve(deq,x(t)); deq := (m[M℄ + m[U℄) · di(x(t),t,t) + · di(x(t),t) + d · x(t) + m[U℄ · r · ω2 · os(omega · t) = 0;

Symboli omputations in modern edu ation

Fig. 7.

155

The surfa es of the revolution modeling

mU

l mM

r

x(t)

ωt

d/2

Fig. 8.

c

d/2

The me hani al vibrating system

Fig. 9.

Parametri display of the amplitudes for the vibrating system

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M. Kami«ski

This example is so easy that no maplet is ne essary to obtain the solution here. An appli ation of the symboli pa kages to tea h how to solve the simple dierential equations (with the well known analyti al solutions) automati ally gives the students opportunity to appre iate the role of the boundary and/or initial onditions on the overall solution [4℄. The omputer with its symboli software takes momentarily a role of the tea her saying by error ommands, what is orre t and what is wrong and whether the sequen e of ommands is well-stru tured or not. Sin e a huge number of su h equations to be taught within a traditional ourse of me hani al engineering, the appli ation of symboli omputing enables for sure (1) deeper understanding of various me hani al and physi al phenomena, (2) better study of inuen e of all parameters on the stru tural response, (3) noti ation of the similarities between dierent phenomena des ribed by the same dierential equations and (4) re ognition of the system properties from its response visualization. The last example deals with the dynami al problem, the so- alled Van der Pol os illator  the user see the equation mathemati al form and has some windows to modify the oe ients and redisplay the nal solution (Fig. 10). This example ombines the di ulty of a solution for the dierential equations system (usually even a single ordinary dierential equation is not easy) together with almost impossible imagination how those os illations may look like. Contrary to the partial dierential equation from the next se tion, the nal solution is not easily available in the analyti al form here, so that it needs mu h more training by slight and systemati modi ations of the initial parameters, their inuen e on the nal display and its physi al interpretation.

Fig. 10.

The surfa es of the revolution modeling

This os illator is just the illustration only how we an use the system MAPLE to make a solution and its visualization for dierent more ompli ated ODEs, PDEs and their systems appearing in dierent models and algorithms like theory of ele troni ir uits, various oupled problems in physi s or so. It is not a se ret that the solution to some of them need deeper mathemati al knowledge and the students after multiple trials with no su

ess, even with properly implemented equation, may give up or/and to fall into the on lusion that the MAPLE annot solve this problem. Taking into

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a

ount this di ulty, the software providers reated some spe ial automati adviser to re ognize the equation and its solution method. The separate problem is that some of those equations annot be solved analyti ally, so that we need to propose in our s ript some numeri al te hnique demanding the additional input parameters. Therefore, ea h espe ially omputational tool whi h an simplify this problem, making by the way its reanalysis very attra tive, is wel ome at most during the a ademi edu ation. Let us noti e at this point that the symboli implementation of those equation inuen es systemati ally the edu ation methods for various ourses provided before using the regular lasses and now  moved at all to the omputer labs. The programmers working now with the lassi al languages (like FORTRAN, for example) may in orporate their subroutines into the symboli environment, so that they an supervise the program ow, to use e iently the interoperability options and, espe ially, to employ the visualization options for their, even pure algebrai , operations' large lists. Only the few years before we all needed to know the separate programming language typi al for the symboli environment but now it is quite unne essary, be ause of the i ons' groups as well as the very extended trees of the sub-options built into the main system window. Finally, let us note that the symboli programs grow in parallel to the hardware, so that even more advan ed problems solutions done by MAPLE do not need the very powerful omputers. So that, those systems may be used widely  also by the students on their notebooks during, before and/or after the le tures in a library (as the book examples veri ation). Considering above, the symboli omputation programs may be the very e ient edu ational tools implemented both in the university omputer labs and at home  for a distant learning. Their extended usage is observed and grows naturally in all those subje ts, where mathemati al equations do appear and the reason is two-fold here: (1) there exist the appli ations enters and the internet well-do umented user's manuals, where you an nd a lot of the ready-to-use problems solved before and ommented su iently by their authors; (2) the user does not really need to know most of the mathemati al methods leading to his parti ular solution. This user needs to use the proper ommand (or the ommands sequen e in a worse s enario), sometimes even the appli ation name and then just li k simply the ENTER button or the additional i on. The exponential growth of the users may be explained taking into a

ount a qualitative progress of the numeri al methods in the applied s ien es. It is widely known, that almost the few years ago the methods available were traditionally divided into analyti al and numeri al approa hes, where the last group was adequate to dis rete algorithms rather like the nite dieren es, nite or boundary elements than the expli it solution of some ODEs system made without any dis retization (just for the equations transformed to the editor as they appear on the paper sheet). Today we an observe that all analyti al methods have been onsumed by the symboli environments, so that for simpler problems no one wants to engage quite separate dis retization and the omputer program when the solution and visualization is easy in the CAS programs (and may be at on e slightly modied to he k the parameter sensitivity of an answer). On the other hand, the a ademi software for the s ienti omputations written in lower level languages may be easily in orporated here into the ex iting graphi al platform, so that we theoreti ally an have everything in a single omputer program realized a

ording to our philosophy. Sin e the leading omputer software ompanies systemati ally develop their systems autonomously, our `ugly' omputational routines written long ago may get a year after year the new pa kaging and the intera tion options automati ally. The small group of the hybrid analyti al-numeri al methods in luding initially symboli algebra may be ome in the future the largest lass on the market in luding most of the

omputational software. The main role of the symboli programs is to enable the students to dis over the existing knowledge in an easier and de isively more attra tive way than before. The very important tools are (1) a programming language very lose to the natural English (some other versions like German, Fren h or Japanese also available), (2) the very extended and intera tive `Help' rea ting for the words similar to some internal pro edures with a large variety of the ready-to-use examples for

opy-paste usage, (3) graphi al environment  the users may hange the domain of both 2D and

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M. Kami«ski

3D plots and animations, multiple fun tions or parametri families may be plotted for the same intervals of the independent variables. Therefore, the students may distinguish between dierent mathemati al models for the same phenomenon or the same mathemati al model but with slightly modied parameters. However, the key feature of the symboli programs lies undoubtedly in the multilevel stru ture of mathemati al operations, the solution veri ation and dynami al denition of the types and names. It is visible at most for the dierential equations and their systems, whi h are ommonly used in some basi s ien es as well as almost the entire engineering. They are at the same time too

ompli ated to be remembered even by better students. The symboli omputing programs give the brand new opportunity for our students  an opportunity to dis over the new fa ts and the resear h ndings they annot nd at a single or even at any available book. Using mathemati al denitions of some quantities and some symboli operations they an derive (with the tea her's aid) some more advan ed properties of these quantities or the properties of their fun tions (like produ ts, the powers or so). It needs essentially more mathemati al ba kground and intuition but an opportunity to dis over something new is the very ex iting by itself, so that the edu ational pro ess is done by the way only (by the way for the students, of

ourse). The aspe t of a time spent on the edu ation is also very important here  having su h a omputational tool there is no need to (1) derive slowly and systemati ally everything on the bla kboard, (2) re ommend the students the additional books in the libraries, (3) prepare the new and extend the existing presentations, whi h may be de isive onsidering the still de reasing number of the le turing and onta ting hours with the students everywhere. Finally, let us note that the MAPLE and related systems oer today the opportunity to prepare the entire le tures and even books into them. We may use to this purpose the automati renumbering of the equations appearing into the text as well as automati saving in the *.html format (some examples an be downloaded from the appli ations enters  http://www.maplesoft. om). There is also the huge library of the mathemati al symbols, Greek letters as well as the other graphi al tools to prepare the ex ellent presentation.

5. THE EXAMPLE FOR THE PHD STUDENTS

As the edu ational example for the symboli omputations during the PhD ourses (related to probabilisti methods and/or reliability analysis) we propose the following transient heat ow problem governed by the following ordinary se ond order dierential Fourier equation [2, 7℄,

∂ ρcT˙ − ∂x

  ∂T k − g = 0; ∂x

x ∈ Ω;

τ ∈ [0, ∞),

(1)

where c = c(T ) means the random heat apa ity hara terizing some domain Ω , ρ = ρ(T ) is the random density of the material ontained in Ω , k = k(T ) stands for the random thermal ondu tivity, while g = g(T ) is the deterministi rate of heat generated per unit volume; the variables T and τ denote temperature eld values and time, respe tively. Although it is possible to solve this problem using the rude Monte-Carlo simulation te hnique, whi h an be easily implemented in MAPLE using the internal statisti al pro edures, we use the sto hasti perturbation te hnique based on the expansion of all fun tions and parameters in this equation into the Taylor series with random oe ients. The edu ational goal is to show how we an solve some transient problem with random parameters using the method, where the randomness is inserted using in a quite deterministi manner using the random quantities probabilisti moments instead of the large statisti al populations. This equation should fulll the boundary onditions of the ∂Ω being a ontinuous and su iently smooth ontour bounding the Ω region. The boundary onditions proposed for Eq. (1) are deterministi and they are dened as follows:

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159

1. temperature (essential) boundary onditions

T = Tˆ;

(2)

x ∈ ∂ΩT ,

and for ∂Ωq part of the total ∂Ω ; 2. heat ux (natural) boundary onditions

∂T = qˆ; ∂n

(3)

x ∈ ∂Ωq ,

where ∂ΩT ∪ ∂Ωq = ∂Ω and ∂ΩT ∩ ∂Ωq = {∅}; 3. initial onditions

xi ∈ Ω ,

T0 = T (xi ; 0);

τ = 0.

(4)

It is possible to dis uss the inuen e of randomness for the initial and boundary onditions later on, using the same example, to demonstrate the students an essential dieren e between the pro ess, where the solution is deterministi ally given as 0 and the situation, when its expe ted value equals 0 and higher moments do not vanish. Let us suppose further that heat ondu tivity and volumetri heat apa ity are un orrelated Gaussian random elds dened uniquely by their two probabilisti moments as follows

E[k], E[ρc] and

Var (k), Var (ρc),

(5)

whi h ompletes the probabilisti des ription of physi al properties of the omposite onstituents. This assumption is not so natural for this moment as for the statisti al analysis, where it is ne essary to generate numeri ally the distributions for those parameters. We will provide a symboli solution to this problem in the form of the expe ted values and standard deviations of temperature eld for this heated stru ture. Therefore, let us denote the orresponding random ve tor of the problem by b(x), with probability density fun tions p(b), so that the mth order entral probabilisti moment is given by +∞ Z µm (b) = (br − E[br ])m p(b) db.

(6)

−∞

As it is known, the basi idea of the sto hasti perturbation approa h follows the lassi al perturbation expansion idea and is based on approximation of all input variables and the state fun tions of the problem via trun ated Taylor series about their spatial expe tations in terms of a parameter ε > 0. For example, in the ase of random heat ondu tivity, the nth order trun ated expansion may be written as [6℄

1 1 ∂ nk k = k 0 + εk ,b ∆b + ε2 k ,bb ∆b∆b + · · · + εn n (∆b)n 2 n! ∂b

(7)

where

ε∆b = ε(b − b0 )

(8)

is the rst variation of b about its expe ted value and, similarly,

ε2 ∆b∆b = ε2 (b − b0 )2

(9)

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M. Kami«ski

is the se ond variation of b about its expe ted value, where nth order variation an be expressed a

ordingly. We an use the internal fun tion of MAPLE to provide this expansion automati ally. The temperature may be expanded a

ordingly as

1 1 ∂nT T = T 0 + εT ,b ∆b + ε2 T ,bb (∆b)2 + · · · + εn n (∆b)n . (10) 2 n! ∂b Traditionally, the sto hasti perturbation approa h for all the physi al problems is entered by the respe tive perturbed equations of the 0th , 1st and su

essively higher orders being a modi ation of the variational integral formulation. Hen e, there holds 1. one zeroth-order partial dierential equation  Z  Z Z 0 ∂T 0 0 ˙0 0 ∂T 0 ρ c T δT + k δ dΩ = qˆ δT d(∂Ω ) + g 0 δT dΩ , ∂x ∂x ∂Ωq

Ω

(11)

Ω

2. nth order higher order partial dierential equation     Z n−p n   X X n (p) n − p (m) ˙ (n−p−m)  δT  ρ c T dΩ p m Ω

p=0

+

Z

Ω

m=0

δ

Z Z n   ∂T X n (p) ∂T (n−p) k dΩ = δT (g)(n) dΩ + δT (ˆ q )(n) d(∂Ω ). ∂x p ∂x p=0

Ω

(12)

∂Ωq

The symboli omputations program may be very useful at this moment also for automati derivation of the in reasing order equations (formation of those equations and olle tion of the same order omponents). Having solved those equations for T 0 and their higher orders respe tively, we derive the expressions for the expe ted values and the other moments of the temperature eld. In order to al ulate the expe ted values and higher order probabilisti moments of temperature T (b; t), the same Taylor expansion is employed to the denitions of probabilisti moments +∞ +∞  Z Z 1 n ,n 0 ,b n E[T (t, b); b] = T (b) p(b) db = T + εT ∆b + · · · + ε T (∆b) p(b) db. n! −∞

(13)

−∞

Sin e the symboli integration is available in MAPLE, we an do that in two dierent ways  by a dire t integration a

ording to this formula, and, on the other hand, by employing this perturbation under the integral and using the denitions holding true for Gaussian variables (for the standard deviations σ )

µ2k+1 (b) = 0,

µ2k (b) = (2k − 1)! σ 2k (b);

k ≤ m.

(14)

Using this extension of the random output, a desired e ien y of the expe ted values an be a hieved by the appropriate hoi e of m and ε orresponding to the input probability density fun tion (PDF) type, probabilisti moment interrelations, a

eptable error of the omputations, et .; this hoi e an be made by omparative studies with Monte-Carlo simulations or theoreti al results obtained by dire t (i.e. symboli ) integration. On e more the same symboli omputations pa kage seems to be the best omputational tool be ause we an operate the additional polynomial forms dire tly; the same general implementation in some programming language ould be extremely di ult and ould be the problem itself. The omputational part is provided using the following data: E[k] = 1.0 BTU/(se in ◦ F), E[ρc] = 1.0 BTU/(in3 ◦ F), where the deterministi result [2℄ is provided using those expe tations instead of their deterministi ounterparts (  1    ) 2 κt 2 x2 x x √ T (x, t) = exp − − erfc , (15) k π 4κt 2 2 κt

Symboli omputations in modern edu ation

Fig. 11.

Fig. 12.

Temperature distribution along the heated rod in time

The expe ted values of temperatures in the heated rod 2

161

Fig. 13.

The standard deviations of temperatures in the heated rod

k where κ = ρc = 1.0 in sec is further used for the probabilisti modeling. Thanks to the appli ation of the symboli omputations pa kage it is possible to visualize the deterministi solution (Fig. 11) with respe t to the spatial and temporal oordinates at the same time, whi h is obtained automati ally at on e. We may point out here that this parti ular heating pro ess, against the homogeneous hara ter of the rod, has essentially dierent hara ter at the opposite ends of this stru ture  let us ompare at least x = 0 with x = 3. It is also possible to make an animation showing dynami al temperature

hanges within the heated rod as well as to arry out the parametri studies by a olle tion of dierent surfa es on a single graph resulting from dierent ombinations of heat ondu tivity and volumetri heat apa ity. Furthermore, we investigate the probabilisti problem, where the expe ted values and standard deviations are visualized. As it is lear from Figs. 12 and 13 reporting those omputations, the expe ted values during the entire heating pro ess are larger than the orresponding deterministi values; the oe ients of variation of the temperature are equal more less 0.1, whi h an be

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re overed dividing the results from Fig. 13 by those olle ted in Fig. 12, respe tively. This study after small modi ations may be dire tly useful in the reliability studies of su h stru tures and for all those ases, where the solution of the relevant state dierential equation may be symboli ally obtained. A plenty of other symboli omputations appli ations in homogenization, strength, plasti ity, fra ture and fatigue of omposite materials may be found in [5℄. Finally, let us note that the symboli omputations based Ph.D. ourses may be provided not only for the analyti al methods and analyti al solutions to more popular ordinary and partial differential equations. The linear algebra library give the very onvenient tools for an illustration of the entire dis retization by the modern dis rete omputational methods like the Finite Element Method for instan e. It is possible to show step by step (a) determination of the elemental stiness matri es, (b) omposition of the global stiness matrix, ( ) imposition of the boundary onditions, (d) solution of the algebrai equations system, (e) visualization of the nal displa ements, strains and stresses, (f) provide the parametri studies of those fun tions and (g) arry out the sensitivity studies using both analyti al as well as the Dire t Dierentiation Method [9℄ or analogously implement the Sto hasti Finite Element Method [6℄. The edu ational value of the symboli programs is signi antly higher than the ommer ial or even a ademi FEM systems be ause (a) all operations are visible (unlike in ommer ial FEM realizations that must be treated as the bla k box modelers), (b) it is easier to explain the ru ial moments and tri ks of the FEM on the single MAPLE s ript than by the several routines written in any programming language that annot be displayed at on e on the s reen. The students are fo used on the really important problems of the FEM itself instead of distribution of their attention between dierent les, the programming language semanti s, algebrai operations algorithms and the nite elements idea. Of ourse, we an solve the problems with the seriously limited number of degrees of freedom only but, thanks to the MAPLE interoperability with FORTRAN sour es, some operations may be provided symboli ally here, while the rest is left for the existing (or available in the ri h literature) a ademi programs [8, 10℄. The symboli software is the best opportunity to make a visualization of the shape fun tions for any nite element available as well as their partial derivatives, to explain graphi ally the FEM dis retization idea by olle ting all nite elements and their displa ements elds as the pat hes on a single graph et . This software appli ation in demonstration of the elastodynami s or nonlinear problems solutions are undoubtedly the easiest and the fastest way to understand the FEM te hniques. The same observations hold true for the other dis rete omputer methods like the Finite Dieren e Method, the Boundary Element Method, the Finite Volume Method et . as well as their sto hasti extensions. 6. FINAL REMARKS

1. As it was dis ussed and presented here, the symboli omputing programs give a good, reliable and modern alternative to the traditional methods for many ourses provided at the high s hools and universities. Su

essfully for the tea hers and le turers, they still evolve to enlarge an edu ational impa t on the students and to fasten the omputational part of a given problem solution. Now, the symboli programs remain edu ationally extremely worthy be ause, thanks to their visual attra tiveness omparable to many games and the internal language very lose to the natural (English standard), they impose an algorithmi thinking of the users and, on the other hand, an opportunity to implement many important problems from the basi s ien es and almost all engineering problems. A

ording to the time limitations for all omputer s ien e lab lasses and apabilities of the CAS systems large enough for several a ademi ourses, the examples solved for and by the students must be well prepared and spe i ally hosen to nd the most ru ial features, options and the solution methods. 2. It is important, as it was dis ussed above, that a le turer of the omputer s ien e should be a supervisor of the students not only for the lasses but also of their general a tivity on the

omputers  to avoid the usage of the unveried and improper knowledge sour es existing in the internet and reated sometimes by the students to speed up their edu ational su

esses. On

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the other hand, the a ademi tea hers should instantaneously monitor playing games by the students to prevent or even to limit a mental inuen e of the games with multiple lives, where they are taught that ea h problem (does not matter virtual or real) has several solution trials. It is in a fundamental ontradi tion with the engineering reliability on ept, where ea h stru ture or system has a single servi e life with no renewal options. 3. It be omes apparent last years that the symboli software is so popular, it has so universal

hara ter and so many possible and the well-do umented appli ations that it appears on all types of the omputers, operating systems and even various portable devi es. Its popularity resulting from the highest level programming language may even lead in some future to an integration of the symboli pa kages with the operating systems, so that the new omputers will be equipped (like portable devi es) with the modied versions of the software belonging today to the CAS lass. Nevertheless, its role in edu ation should instantaneously in reases in all a ademi ourses, also for the high s hool and maybe even elementary ourses. REFERENCES

[1℄ I.N. Bronstein et al.Tas henbu h der Mathematik (in Polish). Polish S i. Publ., Warsaw, 2004. [2℄ H.S. Carlsaw, J.C. Jaeger. Condu tion of Heat in Solids. Oxford Univ. Press, London, 1959. [3℄ B.W. Char et al.First Leaves: A Tutorial Introdu tion to Maple V. Springer-Verlag, Waterloo Maple Publishing, 1992. [4℄ C.H. Edwards, D.E. Penney. Elementary Dierential Equations with Boundary Value Problems, 4th edition. Prenti e Hall, New Jersey, 2000. [5℄ M. Kami«ski. Computational Me hani s of Composite Materials. Springer-Verlag, LondonNew York, 2005. [6℄ M. Kami«ski. Generalized perturbation-based sto hasti nite element method in elastostati s. Computers and Stru tures, 85(10): 586594, 2007. [7℄ M. Kami«ski. Symboli omputations in s ien e and engineering. In: E. Mastorakis et al., eds., Pro . 12th WSEAS CSCC Multi onferen e on Computers, pp. 10251031, Heraklion, Gree e, WSEAS Press, 2008. [8℄ M. Kleiber. Introdu tion to the Finite Element Method (in Polish). Polish S i. Publ., Warsaw, 1986. [9℄ M. Kleiber et al.Parameter Sensitivity in Nonlinear Me hani s. Wiley, New York, 1997. [10℄ O.C. Zienkiewi z. The Finite Element Method in Engineering S ien e, 2nd edition. M GrawHill, London, 1971.