In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory during its applications to general relativity and differential geometry in the early twentieth century. The basis of modern tensor analysis was developed by Bernhard Riemann in a paper from 1861. A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows an efficient expression of such tensor fields and operations. While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. The number of indices equals the degree (or order) of the tensor. For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.
Applications Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Élie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
Notation for indices
Basis-related distinctions
Space and time coordinates Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:
The lowercase Latin alphabet a, b, c, ... is used to indicate restriction to 3-dimensional Euclidean space, which take values 1, 2, 3 for the spatial components; and the time-like element, indicated by 0, is shown separately. The lowercase Greek alphabet α, β, γ, ... is used for 4-dimensional spacetime, which typically take values 0 for time components and 1, 2, 3 for the spatial components. Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Otherwise, in general mathematical contexts, any symbols can be used for the indices, generally running over all dimensions of the vector space.
Coordinate and index notation The author(s) will usually make it clear whether a subscript is intended as an index or as a label. For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector A = (A1, A2, A3) = (Ax, Ay, Az) shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression Ai, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are only labels, not variables. In the context of spacetime, the index value 0 conventionally corresponds to the label t.
Reference to basis Indices themselves may be labelled using diacritic-like symbols, such as a hat (ˆ), bar (¯), tilde (˜), or prime (′) as in:
X ϕ ^ , Y λ ¯ , Z η ~ , T μ ′ {\displaystyle X_{\hat {\phi }}\,,Y_{\bar {\lambda }}\,,Z_{\tilde {\eta }}\,,T_{\mu '}}
to denote a possibly different basis for that index. An example is in Lorentz transformations from one frame of reference to another, where one frame could be unprimed and the other primed, as in:
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