masfak tensor andj and rask 2003

8/12/2019 Masfak Tensor Andj and Rask 2003

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The 6-th International Conference of TensorSociety on Differential Geometry and itsApplicationsandMathematical Mathematical Foundation ofInformation Sciences and its Applications

On occasion of the anniversary ofAkitsugu Kawaguchis 100 years birthwho is the founder of Tensor society

August 5 9, 2002.Tsukuba Tokai University


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Dedicated to our professors which I meet alongmy way ofTensor Calculus Application in Mechanics

* the anniversary of Akitsugu Kawaguchis 100years Birth who is the Founder of Tensor societyand

* academician Tatomir Andjeli} (1903-1993) andprof. dr Danilo Ra{kovi} (1910-1985), Yugoslavscientists


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In the first part of my lecture I will speak of my

meeting with tensor calculus, which happened at 1970.,when I was, but a young engineer by education, and amathematician by soul, and a postgraduate student of

engineering mechanics at the newly founded NisUniversity (in 1965), with published papers in Analyticalmechanics and with the first monography published in

Serbian titled Tensor Calculus written byacademician Tatomir Aneli in 1966 and with themanuscript of my professor of mechanics,elastodynamics and theory of vibration and Theory of

Elasticity, prof. Dr Ing. Math. Danilo P. Rakovi,which practically introduced me to the application of thetensor calculus in discrete and continuous mechanics.


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Academician Aneli introduced tensorcalculus into the studies of mathematics andmechanics at the Natural and Mathematical

Sciences High School in Belgrade (1946),and professor Danilo P. Rakovi,introduced tensor calculus into the studies of

mechanics at the technical sciences HighSchools in Serbia. He brought the high

mathematical level of tensor calculus closerto students of technical sciences andmechanical engineers.


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One of academician Anelis importantmonographs was published in German bySpringer-Werlag [Aneli, T. P., (1968),Tensorrechnung nebst Anwendungen(Mathematische Hilfsmittel des Ingenieurs)Springer-Verlag, Berlin, 1968.] In this part of mylecture I will present the content of researchresults of important Serbian scientists (TatomirP.Anjeli (1903-1993), Danilo P. Rakovi (1910-1985), Rastko Stojanovi (1932 -1972) and VeljkoVujii (1929)) which deal with the application oftensor calculus in mechanics.


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Thanks to my professors who thought me the basics of

tensor calculus, aside from my work in the area of non-linear mechanics which was influenced by the works ofacademician Yu.A. Mitropolsky form Kiev, I started tothink about engineering problems of nonlinear dynamicsin a tensor way. That is the way that brought my firstpaper in this area to life. It was titled 148. Hedrih(Stevanovi), K.: (1993), The mass moment vectors at n-

dimensional coordinate system, Tensor, Japan,Vol 54 (1993), pp. 83-87. which I sent to the International Conference ofTensor Society on Differential Geometry and its

Applications 1992 Bucharest, on occasion of the 90-thAnniversary of Akitsugu Kawaguchis birth who is thefounder. Unfortunately I did not present it myself because,at that time, I was obliged to participate in the work of the

ICTAM92 Haifa of IUTAM in Haifa.


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The publishing of my, above mentioned, paper in theTensor journal brought me more inspiration and Itook it as support from the Tensor Society, whichhappened during the scientific blockade of

Yugoslavia, so that I, apart from my basic researchin the area of nonlinear mechanics, continued tothink in the tensor way which led to the creation of

several papers published under the names listed bythe list which apply tensor calculus in the mechanicsof discrete and continual systems.


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Paper titled: Hedrih (Stevanovi}), K. (1999),Comments to the Fundamental Theorem and

Extension of the Rotation Tensor of theDeformable Body, Tensor, N.S., vol. 61, No 3(1999), pp. 290-303. (publishing process in 2002)I wrote at the time when Nato bombs were fallingon my city Ni{ and University, in the period fromMarch to June 1999. Acceptance of this paper forpublishing in the tensor society journal wasanother encouragement for me to continue mywork in the area of tensor calculus application inmechanics.


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The founder of the Japanese scientific Tensor Society, thehonorable scientist Akitsugu Kawaguchi. together with thecurrent members-scientists of the society headed by you,honorable professor dr Tomoaki Kawaguchi made me greathonor by inviting me to hold an invited lecture at yourSymposium which is well-known and respected throughoutthe world. I also see this as a close meeting of the Japaneseand Yugoslav scientists and I am happy for that. Myparticipation in the work of the symposium will be mypayment of respect to the founders of the Tensor society inJapan, but at the same time I would like to give great thanksto my professors Tatomir P.Adjeli (1903-1993) and DaniloP. Ra{kovi} (1910-1985) who thought me tensor calculus andthat remarkable language for mutual comprehension ofmathematics, mechanics and mechanical engineering dynamics.


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Professor Dr.AkitsuguKawaguchi(Pictured on his seventiethbirthday, 8tu April 1972)

(April 8, 1902, KumamotoPrefecture, Kyushu Region)


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Profesor dr Ing. dipl. Math.Danilo P.

Ra{kovi}(1910-1985)Yugoslavia


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Academician

TATOMIR

P.ANDJELI

(1903 - 1993)Yugoslavia


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Academician Tatomir P. Andjeli} andProf. dr Ing Dipl. Math Danilo P. Ra{kovi} with Colleagues


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Professor Dr. Akitsugu KawaguchiTensor N.S. Vol. 24(1972)Professor Akitsugu Kawaguchi was born onApril 8, 1902, in Kumamoto Prefecture, KyushuRegion. From 1922 to 1928 he had his schoolingat Tohoku University in Sendai (Tohoku Regionin Japan) where he did his graduate work takingBachelor of Science degree in 1925. In 1931 hereceived the doctorate of science from the someUniversity.


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In 1928 hi was appointed Acting Member of the newly-

organized faculty of Sciences of Hokkaido University inSapporo, in order to organize the development ofMathematics in the some Faculty. From 1928 to 1930 at the

request of the Government of Japan he visited Germany,Italy and America in order to study Mathematics, especiallyDifferential geometry. On ther foundation of the faculty ofscience at the Hokkaido University in 1930 he was appointed

Associate Professor of the same University and acted as thedirector of the Department. Since 1933 hi has been ordinary

professor of the same University, and since 1952 he has held

an additional post. Professor of the Nihon University inTokyo, and organized in 1959 the new department ofMathematics at the College od Science and Engineering ofthe same University.


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From 1953 to 1962 he often traveled overseas forattending the International Congress ofMathematicians or delivering lectures to students

of many universities in the world.

Akitsugu Kawaguchi is the Founder of Tensor

Society andfirst presidentof the Tensor Societysince 1938. Tensor societry is of internationalnature.


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The above stated three trips in abroad gavehim the good and fortunate chances to makewarmer his friendships with many friends inaboard as well as to get personalacquainstances of very many young

mathematicians, physists and mechanists in theworld, and the Tensor Society could get manynew members in virtue of his trips.


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Not only in the field of pure scientificresearch, but also his contribution to

mathematical education in schools in Japan isso great that one cannot disregard it. He keepthe position of the President of the Hokkaido

Association of Mathematical Education morethan ten years and presided the annual meetingof the Association held elsewhere in Hokkaidoevery year in order to guide studies ofmathematical education by school teachers.


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. His greatest merit in the introdution of theconcept motion, consisting of paralleldisplacement, rotatio and reflexion, and similar

transformation in order to understand theproperties of figures (geometry) in place ofEuclid Elements into School-mathematics

from standpoint of modern idea in geometry. This concept was adopted as the standard

one in the Guide-Principle of Education

(Gakusha Shido Yoryo) published by theMinistry of Education in Japan.


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Academician

TATOMIR P. ANDJELI

(1903 - 1993)


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Dr Tatomir P. Andjeli} was professor of the Facultyof Natural Sciences and Mathematics, University of

Belgrade, ful member of Serbian Academy of Sciencesand Arts and International Astronautical Academy inParis. T.P. Andjeli} studied mathematics, physiscs andasronomy which lead to his formation as theoretician of

mechanics and astrodynamics. He published over 100

scientific and university works and papers, among whichare 17 university textbooks. He addressed the problems

of rational mechanics, Reiman geometry, astrodynamics,

numerical methods in mathematics and also historical

and phylosophical problems of mechanics. He was very

active in scientific societies, especially in disseminating

scientific knowledge.


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Dr Tatomir P. Andjeli} was a Dean ofthe Faculty of Natural Sciences and

Mathematics and Director of

Mathematical Institute of the SR of Serbia.He had a long and very fruitful life, that

he was in good healt which enabled him to

be active fir a long time, and that he wrote

his last book

Introduction to Astrodynamicsat the age of 80.


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ProfessorAndjeliwas born on November 11, 1903 in

Beanj, an village with no elementary school, so that hehad to walk five kilometers to school every day. At thebeginning of this century, the Andjelis home, althoughin village, had a rich library and echoed with foreignlanguages occasionally.

Having completed the grammar school in aak, theyoung Andjeli decided to go to Heidelberg, the oldest

and highly German University. At the university ofHeidelberg he enjoyed particular attention by professor

Artur Rosentall, an outstanding teacher and anti-Nazi,

as well as by professor Heinrich Leibmann. Dwelling inBelgrade onward, Andjeli passed his doctoral exam1946. Became an assisteant professor in 1946, associateprofessor in 1951, and full professor in 1956.


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Professor Tatomir P. Andjeli started his careeras the assistant of the well known andexceptionally creative Milutin Milankovi, andlater of pedante and always alert Anton

Bilimovi and young Vjeeslav ardecki .

Professor Andjeli

was a very diligent man,always busy, a man who, in his own way, paidtribute to work and with constant inventivness,

never stopped encouraging and supporting hisstudents in their pursuit of knowledge.


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Linear algebra and matrices were in the focus of

professor Andjeli. The tensors were the secondaffection of professor Andjelis in sciences. Accordingto the compliments, the tensors represented a scientific

field, mathematical discipline, through which hi movedmost familiarly and thoroughly and which he educatedexperts and stimulated scientists. In the period when

only a few monographs concerning tensors had beenwritten, our professor Andjeli published the bookTensor Calculus as early as 1952. This book ranthrough more than five editions, and there are about 25thousand copies of it either in the libraries throughoutour country or in the private possession of variousscientists and experts.


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This could be considered rarity for our

circumstances. In 1970 he held a course in tensorcalculus in Centre international des sciences mecaniquesCISM), Udine (Italy), and those lectures werepublished in English as a separate book: A survey ofTensor calculus, Centre international des sciencesmecaniques (CISM),Udine, 1970, p 137. The prominentpublishing hous Springer Verlag published theMathematical Reference Book for Engineers, forwhich professor Andjeli had written a special entryon tensors.Springer-Verlag [Aneli, T. P., (1968),Tensorrechnung nebst Anwendungen (MathematischeHilfsmittel des Ingenieurs), Springer-Verlag, Berlin,

1968.]


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Aneli, T. P., (1968),

Tensorrechnung nebst Anwendungen(Mathematische Hilfsmittel des Ingenieurs),

Springer-Verlag, Berlin, 1968.

Aneli, T. P.,A survey of Tensor calculus,Centre international des sciencesmecaniques (CISM), Udine, 1970, p 137.


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On the cover page of the university non-standardtextbook a monograph Theory of shell and plate that

academician Andjeli published in 1975, as co-authorwith P.M. Ogibalov, a short review of his scientificbiography could be found:

Dr Tatomir P. Andjeli, professor of the Faculty ofNatural Sciences and Mathematics, University of

Belgrade, full member of the Serbian Academy ofSciences and Arts and international AstronauticalAcademy in Paris.


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Necrology written by corresponding memberBoidar D.

Vujanovi

said: On August 7, 1993., prof. dr TatomirP. Andjeli deceased, full member of the SerbianAcademy of Sciences and Arts and full professor of theFaculty of Natural Sciences andMathematicsProfessor Andjeli was spiritually rich,highly educated and very talented person, whose work atthe University left to scientific audience and to the

Serbian Academy of Sciences and Arts deep trace that ishighly valued in Yugoslav and foreign scientificcircles it is rightly considered that professor Andjeli

is the founder of the Belgrade school of mechanics. Manygeneration of students will remember him as a brilliantprofessor, reliable and wise adviser and director ofscientific projects, doctoral dissertations and studies..


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Only two years before, during the ceremonial opening of ascientific meeting on January 18, 1991, held in honor ofacademician Tatomir P. Andjeli, academicianAleksandar Despi,in his welcome-address, on behalf of the Serbian Academy,expressed the honor and pleasure to open the meeting and

among other, said: This honor and pleasure is great one dueto the fact that I myself belong to the plead of persons to whomprofessor Andjeli, over 50 year ego, gave the very firstknowledge on the power and possibilities of mathematics andmathematical way of thinking. In my first contact with thesecondary school, at that time II (Second) Boys Gymnasiumin Belgrade, the first one who described to me what is the term

space means was young suplent Tatomir P. Andjeli,straight as a pine tree, and for us school pupils, hogh as atower.That forst encounter and the period of two years of meetingprofessor Andjeli, will remain in my memory for ever.


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On the same meeting professor Veljko A. Vujii

emphasized the contribution of Tatomir P. Andjelisstudy on the motion of non-holonomic system in fluid,

importance of scientific and educational activity for thedevelopment of our school of mechanics, and on that

occasion he said: ..In consider it is necessary to emphasizethat, to some extent, the role of school of Suslov is over-stressed with us, and that distinguished place that belong to

Tatomir P. Andjeli concerning the development of ourschool of mechanics is not sufficiently valued. Success of ouranalytical mechanics and continuum mechanics primarilyare based on solid mathematical foundation of Belgrade

school, in majority on functional and tensor analysis anddifferential geometry. Considerable value is to be found inbooks of professor Andjeli dealing with the problems of

matrix, vector and tensor calculus.


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On a commemorative meeting academician Vladan Djordjevisaid thatprofessor Andjelihad a long and very fruitful life, thathe was in good healt which enabled him to be active for a longtime, and that he wrote his last book Introduction toAstrodynamics at the age of 80. He also said that professor

Andjeli

was highly respected and esteemed person, surroundedwith love of his family and his associates. He finished his exposewith professor Andjelis words: In nature and in the worldround us everything is changing and in that constant

transformation development and decline nothing occurs with-out anything that have not existed before and that causedappearance of new things, and concluded saying that todayProfessor Andjeli is no longer among us, but at a number of

faculties of natural sciences and mathematics, technical facultiesand scientific institutes in our country, many scientific workersare engaged in theoretical or applied mechanics and who, directlyor indirectly, came from his school of mechanics


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Academician Andjeli was well known by his wide

culture, great knowledge and extremely good memory.He spoke several languages and had outstandingtalent and sensitivity for the linguistics of our

language. He was also preoccupied with the history ofmathematical and mechanical sciences, as well as ofastronomy, so he held a great number of scientific and

popular lectures in these fields. He resumed thatknowing not only the recent state of sciences, but itshistorical dvelopment as well, is necessary for the true

scientific work. He devoted great attention andenergy to writing about the famous MilutinMilankovi.


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.. Thanks to the fact that he was interested

in the tensor calculus primarily for the purposesof application in mechanics and geometry, prof.

Andjeli succeeded in giving life to this area

through his textbook. His book, which hasunusually large proportion of text in relation toequations for mathematical literature, isrelatively easy to read, easily keeps the attentionof readers, and, in my opinion, still represents anunsurpassed text in this area. Although written in

Serbian it was not unnoticed abroad.


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Academician V. orevi said: And when man in

1957 leaped into space for the first time, prof.Andjelic waited that day with a boy-like curiosity andadmiration. He closely observed each of the

spectacular steps in conquering space, traveled a lotat that time, was very well informed on everythingand was one of the first foreigners who had access to

space research centers, both Russian and American.It is little known that the Apolo 17 crew markingswere based on his idea.At that time he held a number

of lectures on space exploration in the country andmade great effort to bring closer and explain thecomplex laws of nature to common man.


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Poseban uticaj na formiranje i `ivotni putnau~nika Tatomira An|eli}a imali su* veli~anstven i {iroke op{e kultureprofesor, izuzetanMihajlo Petrovi} (1868 - 1943),* znameniti i vi{estruko talentovan ikreativanMilutin Milankovi} (1879 - 1958) i* pedantan i ve~ito budanAnton Bilimovi} (1879 - 1970).


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XI. KRATAK IZVOD IZ SPISKA PUBLIKOVANIH RADOVAIZ OBLASTI I ASTRONOMIJENau~ni radovi

An|eli}, T. P., 1959), ber die Grundlagen derBoscovich, Actes du Symp.intern. R.J. Boscovich,1958, Beograd, Zagreb, Ljubljana, 1959.An|eli}, T. P., 1959), O putawama projektilaka mesecu, Vasiona, 1959.An|eli}, T. P., 1960), Me|uplanetne putaweRad, Beograd, 1960.An|eli}, T. P., 1968), Tensorrechnung nebst

Anwendungen (Mathematische Hilfsmittel desIngenieurs) Springer-Verlag, Berlin, 1968.


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n memory ofProf. Dr. Ing. Dipl. Math.

Danilo P. RAKOVI,full-professor of the Faculties of Mechanical Engineering in

Belgrade, Ni, Kragujevac, Mostar and Novi Sad

Prof. Dr. Ing. Dipl. Math. Danilo P. RAKOVI was the first

head of the Department of Mechanics and Automaticswithin the Faculty of Mechanical Engineering of theUniversity of Ni


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This distinguished scientific figure of exquisite


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This distinguished scientific figure of exquisite

creative energy and inspired enthusiasm, a scholardeeply attached to the Yugoslav and Serbian scientificand cultural heritage and an exquisite pedagogist of highethic principles is in the living memory of many

generations of students whom he taught how to learnand love mechanics, as a basic scientific branch ofmechanical engineering either directly, through his

lectures, or through his various and numerous textbooksand collections of problems which circulate in more than140,000 copies. His disciples and colleagues are glad that

he had the ability to transmit to them his greatenthusiasm permeated with his sincere devotion formechanics and his exquisite scientific eagerness.

Prof Dr Ing Dipl Math Danilo P RAKOVI was born


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Prof. Dr. Ing. Dipl. Math. Danilo P. RAKOVI was born

in 1910 in Uice. Upon completing elementary school and sixgrades of high school, he graduated from the MilitaryAcademy in 1930. As an engineering military officer heenrolled in the Department of Mechanical-Electrical

Engineering of the Faculty of Engineering in Belgrade in1933. Having graduated in 1938, he enrolled in theDepartment of Theoretical Mathematics of the Faculty ofPhilosophy and graduated in 1941. As a graduate mechanicalengineer he was appointed assistant section head of theMilitary-Technical Institute in aak. He remained on thatposition during 1941. In 1942 he was appointed assistant of

the Faculty of Engineering in Belgrade where he earned hisdoctor's degree in 1944 upon the defense of the thesis entitled:"Tangential strains of normally profiled beams".

His university career took the following course: in


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His university career took the following course: inJune 1946 he was appointed assistant professor andlecturer; 1949 - associate professor; assistant professor ofmechanics at the Faculty of Engineering; 1951 - associateprofessor; 1957 - full professor of mechanics.

Prof. Dr. Ing. Dipl. Math. Danilo P. RAKOVI lecturedmechanics, resistance of materials and oscillation theory at thefaculties of mechanical engineering in Belgrade, Ni, Kragujevac,Novi Sad and Mostar, as well as in the Faculty of Science in

Belgrade, Faculty of Philosophy in Novi Sad, Faculty of Electronicsin Ni{ and at the Military-Technical College in Belgrade. Moredetails on the research work of professor RAKOVI can be foundin the Belgrade University Bulletin no. 75 of 1957, issued on theoccasion of his appointment as full professor at the Faculty ofMech. Engineering in Belgrade.

During his university career he was twice elected Vice-Dean


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During his university career he was twice elected Vice Deanof the Faculty of Mechanical Engineering of the BelgradeUniversity. In the Mechanical Engineering Department of theFaculty of Engineering in Ni, he lectured statistics, kinetics,kinematics, dynamics, oscillation theory, resistance of

materials, theory of elasticity, as well as analytical mechanics,theory of nonlinear oscillations and continuum mechanics onthe postgraduate level. He was the first head of Departmentof Mechanics and Automatics of the Faculty of Mechanical

Engineering in Ni. He was an extremely inspired professor,scientist and practitioner much favored among his students,respected by his colleagues both as a professor and an

engineer, because he knew to relate engineering theory andpractice.

Professor Rakovi was a very fertile writer While still in


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Professor Rakovi was a very fertile writer. While still inmilitary service he wrote five professional papers. In theperiod before 1957, when he was appointed full professor, hepublished 26 scholarly papers. As a full professor he wrote 37scientific works published in the scientific journals of the

Serbian Academy of Arts and Sciences, Polish Academy ofScience, German Society for Mechanics ZAMM and in otherforeign journals. He took part in a number of scientificmeetings in the country and abroad. He reviewed papers for

four leading referral journals in the world: Applied MechanicsReview (USA), Mathematical Reviews (USA), Zentrablatt(Germany) and Referativnii urnal (Moscow). ProfessorRakovi was a member of several professional and scientific

societies/associations in the country and abroad, GAMM beingone of them. He initiated the foundation of the YugoslavSociety for Mechanics during 1952.

He wrote a considerable number of university


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ytextbooks which ran through numerous editions. Someof them still hold records as for the number of editionsand copies printed within the group they belong to. Inaddition, he wrote a series of textbooks in the field ofmechanics for secondary technical schools, as well as anumber of chapters in professional technicalhandbooks, mimeographed course materials and

textbooks for post-secondary schools of mechanicalengineering.

He also wrote several textbooks for postgraduatestudies.

We are of the opinion that this occasion should


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We are of the opinion that this occasion should

be used to say a few things about professor Rakovi'sfour-year-long co-operation with the publishing house"Nauna knjiga" concerning the publication of his

university textbooks, collections of problems andvarious other handbooks.

Follows the review of university textbooks andhandbooks written by professor Danilo Rakovi andpublished by "Nauna knjiga":

MECHANICS I (STATICS) - a 18,000-copy printing of firsth di i di i di i b di i


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three editions; II edition 1949, III edition 1950; subsequent editions

1960 (a 3,000-copy printing), 1962, 1964, 1965, 1968, 1971, 1973,1978 (a 3,000-4,000-copy printing per edition)

MECHANICS II (KINEMATICS) - first two editions in a 13,000-

copy printing. Editions: 1947, 1950, 1966.

MECHANICS III (DYNAMICS) - first two editions in 10,000 copies.Editions: 1947, 1956, 1962, 1973.

COLLECTION OF PROBLEMS IN STRENGTH (RESISTANCE)OF MATERIALS - first two editions in a 6,000-copy printing.Editions: 1947, 1955, 1965, 1967, 1971, 1975, 1981, 1985 - a 16,000-

copy printing approximately.

STRENGTH OF MATERIALS - editions: 1955, 1961, 1962, 1965,1967, 1971, 1973, 1977, 1980, 1984 - approximately 25,000 copies.

OSCILLATION THEORY - editions: 1957, 1965 - 6,000


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OSCILLATION THEORY editions: 1957, 1965 6,000

copies.

TABLES OF STRENGTH OF MATERIALS - (ran through13 editions) approximately 40,000 copies.

ESSENTIAL MATRIX CALCULUS -1971 edition, 1,500copies.

ELASTICITY THEORY - 1985 edition, 2,000 copies.

ANALYTICAL MECHANICS

and

TENSOR CALCULUS,both edited by the Faculty of Mechanical Engineering in

Kragujevac.


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Some of his university textbooks were published

by other publishing houses such as "SerbianTextbook Printing Office", "Gradjevinska knjiga","Tehnika knjiga", among which the following ought

to be mentioned:ESSENTIAL THEORY OF MECHANISMS

(1965) 2,000 copies;

MECHANICS II (KINEMATICS), 1966 edition in3,000 copies;COLLECTION OF PROBLEMS IN

MECHANICS III - OSCILLATION THEORY (1969)2,000 copies.

Among professor Rakovi's textbooks for


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Among professor Rakovi s textbooks for

secondary technical schools the followingdeserve to be mentioned:STATICS (a 9,000-copy printing),

KINEMATICS (7,500),DYNAMICS (10,000).

Among the publications for postgraduate studies thefollowing should be mentioned:ANALYTICAL MECHANICS

andTENSOR CALCULUS,both edited by the Faculty of Mechanical Engineeringin Kragujevac.

Most of his university textbooks and publications were at


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y p

the time of their first editions the only professionalliterature in Serbian language in the field. So, hispublications played an important part in spreading theknowledge in the field of technical mechanics among the

students and mechanical and other engineers in Serbia andYugoslavia. It is particularly worth mentioning that he hasinterpreted all his material by the most modern

mathematical apparatus and has illustrated it bynumerous examples from the engineering practice. Manyof the cited university publications are reprinted even nowand are still used by both students of engineering and

engineers themselves.

Although he has left us ten years ago, professor


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g y g , p

Rakovi is still present among the newgenerations of students, as well as among the

engineers through his distinguished textbooks

bearing memory of his merits which have left an

indelible imprint on the development of

mechanical engineering science and practice, as

well as on the formation of many a generation of

university professors. His life and work set an

example serving as a creative impulse to the

forth-coming generations educated at the

University of Ni. He is an everlasting exampleand proof that one's deeds can outlive one's

physical existence by far.


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Professor Rakovi introducedtensor calculus into technical

sciences university study (1963)at faculties of technical sciences

in Serbia


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Mechanics Differential Geometry and applications


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Academician Tatomir AndjeliDanilo RakoviRastko Stojanovi

Acad. ANS Veljko VujiiAcademician SANU Mileva PrvanoviAcademician SANU Boidar Vujanovi

Katica (Stevanovi) Hedrih Neda BokanZoran Drakovi Irena omiJovo Jari Novica Blai

Mirko Vukobrat Nevena Pui

Sradjan VukumiroviJovanka Mici

Yugoslav scientists Tensor Svetislav Mini

Calculus and Applications Ljubica Velimirovi

masfak tensor andj and rask 2003

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