10.1 Secondary sources to work on relativity.7.1 Poincaré's writings in English translation.2.7 Differential equations and mathematical physics.2.3.6 Assessments on Poincaré and relativity.2.3.2 Principle of relativity and Lorentz transformations.The Poincaré group used in physics and mathematics was named after him.Įarly in the 20th century he formulated the Poincaré conjecture that became over time one of the famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman. In 1905, Poincaré first proposed gravitational waves ( ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz in 1905. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. He is also considered to be one of the founders of the field of topology. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime.Īs a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. A construction of this index and the extension of Poincaré–Hopf theorem for vector fields with nonisolated zeroes is outlined in Section 1.1.2 of ( Brasselet, Seade & Suwa 2009) harv error: no target: CITEREFBrasseletSeadeSuwa2009 ( help).Jules Henri Poincaré ( UK: / ˈ p w æ̃ k ɑːr eɪ/, French: ( listen) 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. It is still possible to define the index for a vector field with nonisolated zeroes. To do that, construct a very specific vector field on M using a triangulation of M for which it is clear that the sum of indices is equal to the Euler characteristic. Finally, identify this sum of indices as the Euler characteristic of M. Then use the fact that the degree of a map from the boundary of an n-dimensional manifold to an ( n–1)-dimensional sphere, that can be extended to the whole n-dimensional manifold, is zero. Technique: cut away all zeroes of the vector field with small neighborhoods. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold M. The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of N ε to the ( n–1)-dimensional sphere. In addition, make sure that the extended vector field at the boundary of N ε is directed outwards.ģ. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. Take a small neighborhood of M in that Euclidean space, N ε. Embed M in some high-dimensional Euclidean space. They play an important role in the modern study of both fields.ġ.
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This result may be considered one of the earliest of a whole series of theorems establishing deep relationships between geometric and analytical or physical concepts. But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute integer amounts (known as the index) to the total, and they must all sum to 0. In the special case of a manifold without boundary, this amounts to saying that the integral is 0. It is perhaps as interesting that the proof of this theorem relies heavily on integration, and, in particular, Stokes' theorem, which states that the integral of the exterior derivative of a differential form is equal to the integral of that form over the boundary. Thus, this theorem establishes a deep link between two seemingly unrelated areas of mathematics. The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf. A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0.
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