# Mikhail Ostrogradskii

## Ostrogradskii, Mikhail Vasil’evich

Born Sept. 12 (24), 1801, in the village of Pashennaia, in what is now Poltava Oblast; died Dec. 20, 1861 (Jan. 1, 1862), in Poltava. Russian mathematician. Academician of the Imperial Academy of Sciences (1830).

Ostrogradskii studied at the University of Kharkov from 1816 to 1820 and then attended the lectures of A. Cauchy, P. Laplace, and J. Fourier in Paris between 1822 and 1828. Upon his return to St. Petersburg, he became a professor at the officers’ classes at the Naval Cadet Corps in 1828. He subsequently was a professor at the Institute of Transportation Engineers (from 1830), the Main Pedagogic Institute (1832), the Main Engineering School (1840), and the Main Artillery School (1841).

Ostrogradskii’s work dealt chiefly with mathematical analysis, theoretical mechanics, and mathematical physics. His work in number theory, algebra, and probability theory is also well known. In 1826 he solved the important problem of wave propagation on the surface of a liquid contained in a circular cylinder. In his work on heat propagation in solids and liquids Ostrogradskii obtained the differential equation of heat propagation and simultaneously achieved a number of very important results in mathematical analysis. He developed a formula, Ostrogradskii’s theorem, for converting a volume integral into a surface integral. He introduced the concept of the adjoint differential operator and proved the orthogonality of the eigenfunctions of a self-adjoint operator. He showed that functions could be expanded in series in terms of eigenfunctions and established the localization principle for trigonometric series. The theory of heat propagation in liquids actually was first constructed by Ostrogradskii. He also worked on problems in such areas as the theory of elasticity, celestial mechanics, and the theory of magnetism.

The theorem established by Ostrogradskii in 1828 for converting a volume integral to a surface integral was extended by him in 1834 to *n*-fold integrals. With the help of the theorem, he computed the variation of a multiple integral. In 1836 he derived the rules for the change of variables in double and triple integrals. These rules were published in 1838. He also developed a method—sometimes called the Ostrogradskii method—for integrating rational functions by separating out the rational part of the integral. He obtained important results in the theory of differential equations and in approximation theory.

In theoretical mechanics Ostrogradskii obtained fundamental results connected with the principle of virtual displacements and variational principles of mechanic and solved a number of specific problems. In 1854 he constructed a general collision theory. The general variational principle was in the 1840’s almost simultaneously enunciated for conservative systems by W. Hamilton and for nonconservative systems by Ostrogradskii. In his *Memoir on Differential Equations Pertaining to the Isoperimetric Problem* (1850) Ostrogradskii extended these results to the general isoperimetric problem of the calculus of variations. His work on the motion of spherical projectiles through the air and on the effects of a discharge on a cannon’s carriage was of great interest during his time.

Ostrogradskii was a progressive-minded scientist, who adhered to the viewpoint of scientific materialism. For Ostrogradskii the criterion of the value of mathematical investigations was practicality—the possibility of using the results in practical activity. His research in probability theory was characteristic in this respect. One investigation, for example, laid the foundation for a statistical method of quality control; it was conducted by him with the goal of facilitating the testing of goods supplied to the army.

Ostrogradskii wrote a number of articles for the general reader and pedagogical studies. He also wrote textbooks that, for his time, were of excellent quality. Ostrogradskii was a member of many foreign academies.

### WORKS

*Poln. sobr. trudov*, vols. 1–3. Kiev, 1959–61.

### REFERENCE

Gnedenko, B. V., and I. B. Pogrebysskii.*Mikhail Vasil’evich Ostrogradskii*. Moscow, 1963.