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Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are not exponents, even though they may look as such to the reader only familiar with other parts of mathematics.
Gregorio Ricci-Curbastro (Italian: [ɡreˈɡɔːrjo ˈrittʃi kurˈbastro]; 12 January 1853 – 6 August 1925) was an Italian mathematician. [1] He is most famous as the discoverer of tensor calculus .
In the 1950s Schouten completely rewrote and updated the German version of Ricci-Kalkül and this was translated into English as Ricci Calculus. This covers everything that Schouten considered of value in tensor analysis. This included work on Lie groups and other topics and that had been much developed since the first edition.
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. [1]
Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function.
In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage include propositional calculus , Ricci calculus , calculus of variations , lambda calculus , sequent calculus , and process calculus .
Tensor calculus was developed by Gregorio Ricci-Curbastro between 1887 and 1896, presented in 1892 under the title Absolute differential calculus, [76] and the contemporary usage of "tensor" was stated by Woldemar Voigt in 1898. [77] In 1895, Henri Poincaré published Analysis Situs. [78]