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# Discrete calculus: contributors

From Mathematics Is A Science

Jump to navigationJump to searchCalculus includes exterior calculus, i.e., calculus of differential forms. Discrete Calculus can't work without discrete differential forms, i.e., cochains. It cannot then be separated from the rest of exterior calculus or from algebraic topology. Therefore, the credit for the creation of discrete calculus should first go to the following individuals (roughly 1850 - 1950):

*Hermann Grassmann*: exterior algebra;*Gregorio Ricci-Curbastro, Tullio Levi-Civita*: tensor calculus;*Henri Poincaré*: triangulations (barycentric subdivision, dual triangulation), Poincare duality, Poincare lemma, the first proof of the general Stokes Theorem, and a lot more;*Vito Volterra*: "The first mathematician to have written down Stokes' formula for an arbitrary dimension was probably V. Volterra." -- Dieudonne;*L. E. J. Brouwer*: simplicial approximation;*Élie Cartan, Georges de Rham*: the notion of differential form, the exterior derivative as a coordinate-independent linear operator, exactness/closedness of forms, the de Rham's theorem (the de Rham cohomology is equivalent to the singular cohomology);*Emmy Noether, Heinz Hopf, Leopold Vietoris, Walther Mayer*: modules of chains, the boundary operator, chain complexes;*J.W. Alexander, Solomon Lefschetz, Lev Pontryagin, Andrey Kolmogorov, Norman Steenrod, Eduard Čech*: the early cochain notions;*W. V. D. Hodge*: the Hodge star operator, the Hodge decomposition;*Samuel Eilenberg, Saunders Mac Lane, Norman Steenrod, J.H.C. Whitehead*: the rigorous version of algebraic topology;*Hassler Whitney*: cochains as integrands.