NON-COORDINATE METHOD OF DIFFERENTIALIZATION OF VECTOR FUNCTIONS OF THE SCALARY ARGUMENT
Purpose. The purpose of this paper is to develop a coordinate-free (vector) method for differentiating vector functions of a scalar argument.Methodology is. bBased on the representation of the increment of the vector function of
the scalar argument, which simultaneously changes in direction vector and modulus, as the sum of the increment due to the change in direction and increment due to the change in its modulus, mathematical dependencies are obtained for coordinate-free vector functions of the scalar argument that describe the mechanical phenomena occurring in the fixed and moving reference systems.
Results. In theoretical mechanics, the methods of vector algebra and vector analysis are widely used. Vector calculus due to the compactness and physical clarity of vector formulas has a great advantage over the coordinate method. In modern conditions of higher education, when the number of hours for classroom training is rapidly decreasing, there is an urgent need for studying theoretical mechanics to apply such methods of performing mathematical operations on vector quantities that would convincingly and with little time consumption allow to carry out proofs of certain theoretical positions. As you know, there are two methods for performing mathematical operations on vector quantities: coordinate-free one (or vector), when operations are performed directly on vectors, and coordinate one, in which operations are performed on scalar values that analytically express a vector in a certain coordinate system. The coordinate-free method is more compact and should be used when conducting theoretical studies. Based on the hodograph analysis of the vector function of a scalar argument, the vector which simultaneously changes in direction and modulo it has been established that the increment vector of this function when the scalar argument changes is equal to the geometric sum of the increment of this function due to the change in the direction of its vector and the increment resulting from functions. Based on this, it was found that the vector of the derivative of the function of a scalar argument consists of two components: the vector of the derivative, which characterizes the rate of change in the direction of the vector of the function and the vector of the derivative, which characterizes the change in the modulus of the vector. The paper also considers the case of differentiation by the coordinate method of the scalar argument function, which characterizes a mechanical phenomenon that occurs in a certain frame of reference, which moves relative to other frames of reference, one of which is taken as fixed. It has been established that the absolute derivative of the vector function of the scalar argument in this case depends on the angular velocity of rotation of the moving reference systems and the dependence that characterizes this relationship.
Originality. The developed coordinate-free method for differentiating the vector functions of a scalar argument, the vector of which simultaneously changes in direction and modulo, which is based on the analysis of the properties of the hodograph of the function, is original.
Practical value. ApplicationsThe use of the developed coordinate-free method for differentiating the vector functions of the scalar argument can be used in the educational process and can makes it possible to carry out the proofs of a number of theoretical positions on the subject “Theoretical Mechanics” more convincingly and with less time consumption (see References 14, figures 3.)
Melnikova Н. V., Melnikov Y. B., Melnikova Y. Y. (2006). Basics of vector analysis. Integrals in field theory. Ekaterinburh: GОU VPО «UGTU - UPE» (in Russ.) Retrieved from: https://study.urfu.ru/Aid/Publication/368/1/Melnikovs_Vect_analis.pdf .
Gryniov B. V., Kyrychenko І. К. Ed. Lytvyn О. М. (2008). Vector algebra: textbook for technical universities. Kharkiv: Gymnasium (in Ukr.) Retrieved from: http://www.e-catalog.name/x/x/x?LNG=&Z21ID=&I21DBN=VGPU_PRINT&P21DBN=VGPU&S21STN=1&S21REF=&S21FMT=fullw_print&C21COM=S&S21CNR=&S21P01=0&S21P02=1&S21P03=A=&S21STR=%D0%93%D1%80%D0%B8%D0%BD%D1%8C%D0%BE%D0%B2,%20%D0%91%D0%BE%D1%80%D0%B8%D1%81%20%D0%92%D1%96%D0%BA%D1%82%D0%BE%D1%80%D0%BE%D0%B2%D0%B8%D1%87 .
Galeeva А. А., Chervikov B. G. (2009). Elements of vector analysis. Electronic study guide. Kazan: Kazan National University (in Russ.) Retrieved from: https://old.kpfu.ru/f3/bin_files/el-v!209.pdf .
Malyshev А. I., Maksimova G. М. (2012). Fundamentals of vector and tensor analysis for physicists. Electronic teaching aid. Nizhny Novgorod University (in Russ.) Retrieved from: https://zzapomni.com/urfu-ekaterinburg/melnikova-osnovy-vektornogo-analiza-2006-17977/view .
Kilchevski N. А. (1977). The course of theoretical mechanics, Volume 1 (kinematics, statics, point dynamics). Ed. 2nd. Main edition of the physics and mathematics literature of the publishing house «Nauka», Мoscow (in Russ.) Retrieved from: http://mechmath.ipmnet.ru/lib/?to=search&s_s1=%CA%E8%EB%FC%F7%E5%E2%F1%EA%E8%E9&submit_search=%C8%F1%EA%E0%F2%FC .
Pavlovski М. А., Akinfieva L. Y., Boichuk О. F.; Edited by Pavlovski М. А. (1989). Theoretical mechanics. Statics. Kinematics. Кyiv: Vyshcha shkola, Main Publishing House (in Ukr.) Retrieved from: https://www.studmed.ru/download/pavlovskiy-ma-teoretichna-mehanka-ukr_b09eb95a413.html .
Nikitin N. P. (1990). The course of theoretical mechanics. Textbook. For engineering and instrument-making specialties Universities 5th edition revised and enlarged. Мoscow: Vysshaia shkola (in Russ.) Retrieved from: https://www.for-stydents.ru/teoreticheskaya-mehanika/uchebniki/nikitin-kurs-teoreticheskoy-mehaniki .
Tokar А.М. (2001). Theoretical mechanics. Kinematics. Methods and tasks. Teaching Manual. Кyiv: Lybid (in Ukr.) Retrieved from: http://www.irbis-nbuv.gov.ua/cgi-bin/irbis_nbuv/cgiirbis_64.exe?Z21ID=&I21DBN=EC&P21DBN=EC&S21STN=1&S21REF=10&S21FMT=JwU_B&C21COM=S&S21CNR=20&S21P01=0&S21P02=0&S21P03=U=&S21COLORTERMS=0&S21STR=%D0%92212%20%D1%8F73 .
Butenin N. V., Lunts Y. L., Merkin D. R. (2002). The course of theoretical mechanics. In two volumes. St. Petersburg.: Publishing House «Lan» (in Russ.) Retrieved from: https://www.studmed.ru/download/butenin-kurs-teoreticheskoy-mehaniki-v-dvuh-tomah_663ca50eb60.html .
Pavlovski М. А. (2002). Theoretical mechanics. Kyiv: Tehnika (in Ukr.) Retrieved from: http://btpm.nmu.org.ua/ua/download/Павловський%20М.А.%20Теоретична%20механіка.pdf .
Targ S. М. (2002). Short course in theoretical mechanics. Textbook. 9th Edition. St. Petersburg.: Publishing House «Lan» (in Russ.) Retrieved from: https://fileskachat.com/download/49922_c08fcc3fbfcda612dcd3f982250b5e4d.html .
Bondarenko А. А., Dubinin О. О., Pereiaslavtsev О. М. (2004). Theoretical mechanics. Textbook in 2 Volumes – Volume 1. Static. Kinematics. Кyiv: Znannia (in Ukr.) Retrieved from: http://pdf.lib.vntu.edu.ua/books/2015/Bondarenko_P1_2004_599.pdf .
Drong V. I., Dubinin В. В., Ilin М. М. and others. Under general edition by Kolesnikova К. S. (2005). Theoretical mechanics course. Textbook for high schools. 3rd Edition. Мoscow: Publishing House МGTU named after E.E. Bauman (in Russ.) Retrieved from: http://static.flibusta.site/b.usr/Drong_Kurs-teoreticheskoy-mehaniki.512266.djvu .
Gulivets О. А., Oliinyk S. Y., Markevych G. А. (2017). Vector functions of the scalar argument in the study of point kinematics and solids. Visnyk Cherkaskoho universytetu. Seriia fizyko-matematychni nauky (Bulletin of Cherkasy University. Series of Physics and Mathematics), 1, 138-146. Retrieved from: http://www.google.com.ua/url?url=http://phys-ejournal.cdu.edu.ua/article/download/2397/2468&rct=j&q=&esrc=s&sa=U&ved=0ahUKEwjs2d2K2JvlAhWa7KYKHbMeCJI4ChAWCEEwCQ&usg=AOvVaw0tn8TsJCb7oP8-wIiAV6La.