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.
French mathematician (1865–1963)
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.
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Alternative Names:
Jacques Salomon Hadamard
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... 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 difficulty in the question. ...
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.
... 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.
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.