Editing Mikhail Ostrogradsky

{{short description|Ukrainian-Russian mathematician}}
{{Infobox scientist
| name              = Mikhail Ostrogradsky
| image             = Ostrogradski.jpeg
| caption           = Mikhail Vasilyevich Ostrogradsky
| birth_date        = {{Birth date|1801|9|24|df=y}}
| birth_place       = [[Myrhorod Raion|Pashennaya]], [[Kobelyaksky Uyezd]], [[Poltava Governorate]], [[Russian Empire]]; now [[Kremenchuk Raion]], [[Poltava Oblast]], [[Ukraine]]
| death_date        = {{Death date and age|1862|1|1|1801|9|24|df=y}}
| death_place       = [[Poltava]]
| citizenship       = [[Russian Empire]]
| nationality       = 
| field             = [[Mathematics]]
| alma_mater        = [[University of Kharkiv]], <br/> [[University of Paris]]
| doctoral_students = 
| known_for         = [[Ostrogradsky instability]], <br/> [[Divergence theorem]]
}}

'''Mikhail Vasilyevich Ostrogradsky'''{{efn|also transcribed as '''Ostrogradskiy''' and '''Ostrogradskiĭ'''}} ({{langx|ru|Михаи́л Васи́льевич Острогра́дский}}; 24 September 1801 – 1 January 1862), also known as '''Mykhailo Vasyliovych Ostrohradskyi''' ({{langx|uk|Миха́йло Васи́льович Острогра́дський}}), was a [[Russian Empire|Russian Imperial]]<ref>{{Cite book |last=Kunes |first=Josef |url=https://books.google.com/books?id=_jqUZIUXZBsC&dq=Mikhail+Vasilyevich+Ostrogradsky&pg=PA179 |title=Dimensionless Physical Quantities in Science and Engineering |date=2012-02-13 |publisher=Elsevier |isbn=978-0-12-391458-3 |language=en}}</ref><ref>{{Cite book |last1=Hetnarski |first1=Richard B. |url=https://books.google.com/books?id=18CYMW-CG_gC&dq=Mikhail+Vasilyevich+Ostrogradsky&pg=PA9 |title=The Mathematical Theory of Elasticity, Second Edition |last2=Ignaczak |first2=Józef |date=2010-10-18 |publisher=CRC Press |isbn=978-1-4398-2888-5 |language=en}}</ref> [[mathematician]], [[mechanics|mechanician]], and [[physicist]] of [[Zaporozhian Cossacks]] ancestry.<ref>{{cite web | url=http://www.nbuv.gov.ua/node/4299 | title=Народився Михайло Остроградський, український математик, механік і фізик, розробник методу, правила та формули Остроградського &#124; Національна бібліотека України імені В. І. Вернадського }}</ref><ref>{{MacTutor Biography|id=Ostrogradski}}</ref><ref>{{harvnb|Woodard|2015}}.</ref><ref>[https://bigenc.ru/mathematics/text/2682480 Mikhail Vasilyevich Ostrogradsky (Encyclopedia of Russian Academy of Sciences)]</ref><ref>[https://books.google.com/books?id=_jqUZIUXZBsC&dq=Mikhail+Vasilyevich+Ostrogradsky&pg=PA179 Kunes, Josef. Dimensionless Physical Quantities in Science and Engineering. London — Waltham 2012. P. 179.]</ref><ref>[https://books.google.com/books?id=18CYMW-CG_gC&dq=Mikhail+Vasilyevich+Ostrogradsky&pg=PA9 Hetnarski Richard B., Ignaczak Józef: The Mathematical Theory of Elasticity. USA Taylor and Francis Group, 2011. P. 9.]</ref> Ostrogradsky was a student of [[Timofei Osipovsky]] and is considered to be a disciple of [[Leonhard Euler]], who was known as one of the leading mathematicians of Imperial Russia.

== Life ==
Ostrogradsky was born on 24 September 1801 in the village of Pashennaya (at the time in the [[Poltava Governorate]], [[Russian Empire]], today in [[Kremenchuk Raion]], [[Poltava Oblast]], [[Ukraine]]). From 1816 to 1820, he studied under [[Timofei Fedorovic Osipovsky|Timofei Osipovsky]] (1765–1832) and graduated from the [[Kharkiv University|Imperial University of Kharkov]]. When Osipovsky was suspended on religious grounds in 1820, Ostrogradsky refused to be examined and he never received his Ph.D. degree. From 1822 to 1826, he studied at the [[University of Paris|Sorbonne]] and at the [[Collège de France]] in [[Paris, France]]. In 1828, he returned to the Russian Empire and settled in [[Saint Petersburg]], where he was elected a member of the [[Russian Academy of Science|Academy of Sciences]]. He also became a professor of the [[Military Engineering-Technical University|main military engineering school]] of the Russian Empire.

Ostrogradsky died in [[Poltava]] in 1862, aged 60. The [[Kremenchuk Mykhailo Ostrohradskyi National University]] in [[Kremenchuk]], [[Poltava oblast]], as well as ''Ostrogradsky street'' in [[Poltava]], are named after him.

== Work ==
{{Verification|date=March 2025}}[[File:Coin of Ukraine Ostrograd R.jpg|thumb|right|A 2 [[hryvna]] commemorative coin minted by the [[National Bank of Ukraine]] in 2001.]]

[[File:Меморіальна дошка на будинку, де жив Остроградський.JPG|thumb|left|Commemorative plaque in [[Poltava]] on the last house where Ostrogradsky resided.]]

He worked mainly in the mathematical fields of [[calculus of variations]], [[Integral|integration]] of [[algebraic function]]s, [[number theory]], [[algebra]], [[geometry]], [[probability theory]] and in the fields of [[applied mathematics]], [[mathematical physics]] and [[classical mechanics]]. In the latter, his key contributions are in the [[motion (physics)|motion]] of an [[elasticity (physics)|elastic body]] and the development of methods for integration of the equations of [[dynamics (mechanics)|dynamics]] and [[fluid power]], following up on the works of [[Euler]], [[Joseph Louis Lagrange]], [[Siméon Denis Poisson]] and [[Augustin Louis Cauchy]].

In Russia, his work in these fields was continued by [[Nikolai Brashman|Nikolay Dmitrievich Brashman]] (1796–1866), [[August Davidov|August Yulevich Davidov]] (1823–1885) and especially by [[Nikolai Yegorovich Zhukovsky]] (1847–1921).

[[File:Могила вченого-математика академіка М.В.Остроградського.JPG|thumb|right|Ostrogradsky's grave in the village of [[Pashenivka]], where he was born.]]

Ostrogradsky did not appreciate the work on [[non-Euclidean geometry]] of [[Nikolai Ivanovich Lobachevsky|Nikolai Lobachevsky]] from 1823, and he rejected it, when it was submitted for publication in the Saint Petersburg Academy of Sciences.

Ostrogradsky was a teacher of the children of Emperor [[Nicholas I of Russia|Nicholas I]].<ref>{{Cite web |title=Публикация ННР Некоторые черты из жизни Остроградского |url=http://books.e-heritage.ru/book/10073650 |access-date=2023-02-11 |website=books.e-heritage.ru}}</ref>

===Divergence theorem===
In 1826, Ostrogradsky gave the first general proof of the [[divergence theorem]], which was discovered by [[Lagrange]] in 1762.<ref>For references, see [[Divergence theorem#History]].</ref>  This theorem may be expressed using Ostrogradsky's equation:
: <math> \iiint_V \left( {\partial P\over \partial x} + {\partial Q\over \partial y} + {\partial R\over \partial z} \right) dx \, dy \, dz =
\iint_\Sigma \left( P \cos \lambda + Q \cos \mu + R \cos \nu \right) d\Sigma </math>;
where ''P'', ''Q'', and ''R'' are differentiable functions of ''x'', ''y'', and ''z'' defined on the compact region ''V'' bounded by a smooth closed surface ''Σ''; ''λ'', ''μ'', and ''ν'' are the angles that the outward normal to ''Σ'' makes with the positive ''x'', ''y'', and ''z'' axes respectively; and d''Σ'' is the surface area element on ''Σ''.

===Ostrogradsky's integration method===
His method for integrating [[rational function]]s<ref>{{harvnb|Ostrogradsky|1845a}} and {{harvnb|Ostrogradsky|1845b}}.</ref> is well known. First, we separate the rational part of the integral of a fractional rational function, the sum of the rational part (algebraic fraction) and the transcendental part (with the [[logarithm]] and the [[Inverse trigonometric function|arctangent]]). Second, we determine the rational part without integrating it, and we assign a given integral in Ostrogradsky's form:

:<math> \int {R(x)\over P(x)} \, dx = {T(x)\over S(x)} + \int {X(x)\over Y(x)} \, dx, </math>

where <math>P(x),\, S(x),\, Y(x)</math> are known polynomials of degrees ''p'', ''s'', ''y'' respectively; <math>R(x)</math> is a known polynomial of degree not greater than <math>p - 1</math>; and <math>T(x),\, X(x)</math> are unknown polynomials of degrees not greater than <math>s - 1</math> and <math>y - 1</math> respectively.

Third, <math>S(x)</math> is the greatest common divisor of <math>P(x)</math> and <math>P'(x)</math>. Fourth, the denominator of the remaining integral <math>Y(x)</math> can be calculated from the equation <math>P(x) = S(x)\,Y(x)</math>.

When we differentiate both sides of the equation above, we get:<br/><math> R(x) = T'(x)Y(x) - T(x)H(x) + X(x)S(x) </math>,

where <math>H(x) = {Y(x) S'(x) \over S(x)}</math>.

It can be shown that <math>H(x)</math> is polynomial.

==See also==
*[[Divergence theorem|Gauss-Ostrogradsky theorem]]
*[[Green's theorem]]
*[[Ostrogradsky instability]]

==Notes==
{{notelist}}

==References==
{{reflist}}
*{{Citation
 | last = Ostrogradsky
 | first = M.
 | title = De l'intégration des fractions rationnelles
 | journal = Bulletin de la classe physico-mathématique de l'Académie Impériale des Sciences de Saint-Pétersbourg
 | volume = 4
 | pages = 145–167
 | year = 1845a
 }}.
*{{Citation
 | last = Ostrogradsky
 | first = M.
 | title = De l'intégration des fractions rationnelles (fin)
 | journal = Bulletin de la classe physico-mathématique de l'Académie Impériale des Sciences de Saint-Pétersbourg
 | volume = 4
 | pages = 286–300
 | year = 1845b
 }}.
*{{Cite arXiv
 | last = Woodard
 | first = R.P.
 | eprint = 1506.02210
 | title = The Theorem of Ostrogradsky
 | date = 9 August 2015
 | class = hep-th
 }}

==External links==
*{{MacTutor Biography|id=Ostrogradski}}
*{{cite arXiv |last=Woodard |first=R.P. |author-link= |eprint=1506.02210 |title=The Theorem of Ostrogradsky |class= hep-th|date=9 Aug 2015 |ref=none}}

{{Authority control}}

{{DEFAULTSORT:Ostrogradsky, Mikhail Vasilievich}}
[[Category:1801 births]]
[[Category:1862 deaths]]
[[Category:People from Poltava Oblast]]
[[Category:People from Kobelyaksky Uyezd]]
[[Category:Ukrainian mathematicians]]
[[Category:19th-century mathematicians from the Russian Empire]]
[[Category:Physicists from the Russian Empire]]
[[Category:National University of Kharkiv alumni]]
[[Category:University of Paris alumni]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Full members of the Saint Petersburg Academy of Sciences]]
[[Category:Academic staff of Military Engineering-Technical University]]
[[Category:Privy Councillor (Russian Empire)]]