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Jacques Salomon Hadamard (8 December 1865 – 17 October 1963) was a French mathematician, famous for his research in complex analysis, partial differential equations, differential geometry, and analytic number theory.
- ... In the case of ordinary differential equations, the arbitrary elements being numerical parameters, we have to determine them by an equal number of numerical equations, so that, at least theoretically, the question may be considered as solved, being reduced to ordinary algebra; but for partial differential equations, the arbitrary elements consist of functions, and the problem of their determination may be the chief difficutly in the question. ...
- The true equations which actually lie before us are, therefore, the "boundary problems," each of which consists in determining an unknown function u so as to satisfy:
- (1) an "indefinite" partial differential equation;
- (2) some "definite" boundary conditions.
- Such a problem will be "correctly set" if those accessory conditions are such as to determine one and only solution of the indefinite equation.
- Lectures on Cauchy's problem in linear partial differential equations. Yale University Press. 1923. p. 4.
Four lectures on mathematics, delivered at Columbia University in 1911 (1915)
- In the case of partial differential equations employed in connection with physical problems, their use must be given up in most circumstances, for two reasons: first, it is in general impossible to get the general solution or general integral, and second, it is in general of no use even when it is obtained.
- Just after the discovery of infinitesmal calculus, physicists began by needing only very simple methods of integration, the problems in general reducing to elementary differential equations. But when higher partial differential equations were introduced, the corresponding problems almost immediatelly proved to be far above the level of those which contemporary mathematics could treat.
- We are going to speak of the role of analysis situs in our modern mathematics. This theory is also called the geometry of situation. It is the study of connection between different parts of geometrical configurations which are not altered by any continuouse deformation. For instance, a sphere and a cube are considered as one and the same thing from the point of view of the geometry of situation, because one can be transformed into the other without separating parts, or uniting parts which formerly were separated.
- ... Let a perturbation be produced anywhere, like sound; it is not immediately perceived at every other point. There are then points in space which the action has not reached in any given time. Therefore the wave, in that sense a surface, separates the medium into two portions (regions): the part which is at rest, and the other which is in motion due to the initial vibration. These two portions of space are contiguous. It was only in 1887 that Hugoniot, a French mathematician, who died prematurely, showed what the surface of the wave can be; and even his work was not well known until Duhem pointed out its importance in his work on mathematical physics.
Quotes about Hadamard
- The systematic study of the singularities of analytic functions was begun by Hadamard. In 1901, a very valuable account of his own investigations together with those of other early workers, as Fabry, Leau, LeRoy, Borel and others, was presented by Hadamard in his now classic little book La Série de Taylor et son Prolongement Analytique published in the Collection Scientia (No. 12).
- ... Professor Hadamard concludes that the general pattern of invention, or, as it might also be put, of original work, is three-fold : conscious study, followed by unconscious maturing, which leads in turn to the moment of insight or illumination. Thereupon another period of conscious work ensues, the purpose of which is to achieve a synthesis of several elements: the novel idea, its logically deduced consequences including proof, and the traditional knowledge to which the new item is added.
- Hadamard's investigation, modest and tentative as are its results, seems to me of capital importance in the realm of criticism and cultural history. For what he has done is to show that the human mind tends to behave much the same way whenever it invents, whether in mathematical or in poetic form—a conclusion which does not deny differences of temperament. Our author, on the contrary, is at pains to distinguish among types of mathematical geniuses. He classes them as logical or intuitive, concrete or abstract, yet with enough flexibility to allow for deceptive appearances and for the overlapping of categories. But it is clear in the end that in any process of creation there lurks a mystery—a mystery at least equal to that of thinking itself.