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In mathematics, Ricci calculus is the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–96 and subsequently popularized in a paper written with his pupil Tullio Levi-Civita at the turn of 1900.

Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework (in addition to contributing to the theory), during its applications to general relativity and differential geometry in the early 20th century.

This article summarizes the rules of index notation and manipulation for tensors and tensor fields.

Introduction

Tensors and tensor fields can be manipulated using components, which are labelled by indices. Each index has one possible value per dimension of the underlying vector space. The number of indices equals the order of the tensor.

Conventionally:

• the lowercase Latin alphabet a, b, c... is used for 3-dimensional Euclidean space, which take values 1, 2, 3 for the spatial components.
• the lowercase Greek alphabet α, β, γ... is used for 4-dimensional space-time, which take values 0 for time components and 1, 2, 3 for the spatial components.

Coordinate and index notation

The context should prevent confusion between letters used as indices (which take integer values) and as labels for coordinates.

For example, for 3-D Euclidean space, Cartesian coordinates may be used, and there is a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z in the vector A = (A₁, A₂, A₃) = (A_x, A_y, A_z). In the expression A_i, i is interpreted as an index ranging over the values 1, 2, 3.

Reference to coordinate systems

Sometimes, in the literature indices are labelled for convenience by diacritic-like symbols, such as hats ^, overbars ¯, tildes ~, and primes ′:

${\displaystyle X_{\hat {\phi }}\,,Y_{\bar {\lambda }}\,,Z_{\tilde {\eta }}\,,T_{\mu '}\cdots }$

which usually do not imply an operation: rather they denote another coordinate system specific to the context, for example in Lorentz transformations from one inertial frame to another - one frame is usually unprimed, the other is primed. On the other hand - there exists a convention for using hats and overdots on indices for spinors - see van der Waerden notation.

The following notations do imply operations and are standard in the literature.

Raised and lowered indices

Covariant tensor components

A lower index (subscript) indicates covariance of the components with respect to that index: ${\displaystyle A_{\alpha \beta \gamma \cdots }}$

Contravariant tensor components

An upper index (superscript) indicates contravariance of the components with respect to that index: ${\displaystyle A^{\alpha \beta \gamma \cdots }}$

Mixed-variance tensor components

A tensor may have both upper and lower indices: ${\displaystyle A_{\alpha }{}^{\beta }{}_{\gamma }{}^{\delta \cdots }}$

Summation

Two indices with the same symbol are summed over: ${\displaystyle A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }}$ or ${\displaystyle A^{\alpha }B_{\alpha }\equiv \sum _{\alpha }A^{\alpha }B_{\alpha }\,.}$

The operation of setting two indices equal, and hence an immediate summation, is called tensor contraction:

${\displaystyle A_{\alpha }B^{\beta }\rightarrow A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\,.}$

More than one index may be repeated twice, but only twice and within one term, for example:

${\displaystyle A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,,}$

as for a non-identity

${\displaystyle A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\not \equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,.}$

Groups of indices

If a tensor has a list of indices all raised or lowered, one shorthand is to use a capital letter for the list:

${\displaystyle A_{i_{1}\cdots i_{n}}B^{i_{1}\cdots i_{n}j_{1}\cdots j_{m}}C_{j_{1}\cdots j_{m}}\equiv A_{I}B^{IJ}C_{J}}$

where I = i₁ i₂ ... i_n and J = j₁ j₂ ... j_m.

Sequential summation

Two vertical bars | | around a set of indices (with a contraction):

${\displaystyle A_{|\alpha \beta \gamma |\cdots }B^{\alpha \beta \gamma \cdots }=\sum _{\alpha }\sum _{\beta }\sum _{\gamma }A_{\alpha \beta \gamma \cdots }B^{\alpha \beta \gamma \cdots }}$

denotes the summation in which each preceding index is counted up to (and not including) the value of the next index:

${\displaystyle \alpha <\beta <\gamma \,.}$

Only one group of the repeated set of indices has the vertical bars around them (the other contracted indices do not). More than one group can summed in this way:

${\displaystyle A_{|\alpha \beta \gamma |}{}^{|\delta \epsilon \cdots \lambda |}B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda |\mu \nu \cdots \zeta |}C^{\mu \nu \cdots \zeta }=\sum _{\alpha }\sum _{\beta }\sum _{\gamma }\sum _{\delta }\sum _{\epsilon }\cdots \sum _{\lambda }\sum _{\mu }\sum _{\nu }\cdots \sum _{\zeta }A_{\alpha \beta \gamma }{}^{\delta \epsilon \cdots \lambda }B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda \mu \nu \cdots \zeta }C^{\mu \nu \cdots \zeta }}$

where

${\displaystyle \alpha <\beta <\gamma \,,\quad \delta <\epsilon <\cdots <\lambda \,,\quad \mu <\nu \cdots <\zeta \,.}$

This is useful to prevent over-counting in some summations, when tensors are symmetric or antisymmetric.

Using the convention of capital letters, an underarrow is placed underneath the block of indices:

${\displaystyle A_{\underset {\rightharpoondown }{P}}{}^{\underset {\rightharpoondown }{Q}}B^{P}{}_{Q{\underset {\rightharpoondown }{R}}}C^{R}=\sum _{\underset {\rightharpoondown }{P}}\sum _{\underset {\rightharpoondown }{Q}}\sum _{\underset {\rightharpoondown }{R}}A_{P}{}^{Q}B^{P}{}_{QR}C^{R}}$

where

${\displaystyle {\underset {\rightharpoondown }{P}}=|\alpha \beta \gamma |\,,\quad {\underset {\rightharpoondown }{Q}}=|\delta \epsilon \cdots \lambda |\,,\quad {\underset {\rightharpoondown }{R}}=|\mu \nu \cdots \zeta |}$

Raising and lowering indices

By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:

${\displaystyle B^{\gamma }{}_{\beta \cdots }=g^{\gamma \alpha }A_{\alpha \beta \cdots }}$ and ${\displaystyle A_{\alpha \beta \cdots }=g_{\alpha \gamma }B^{\gamma }{}_{\beta \cdots }}$

The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.

General outlines for index notation and operations

Tensors are equal if and only if every corresponding component is equal, e.g. tensor A equals tensor B if and only if

${\displaystyle A^{\alpha }{}_{\beta \gamma }=B^{\alpha }{}_{\beta \gamma }}$

for all α, β and γ. Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to dimensional analysis).

Indices are the same in every term

The same indices on each side of a tensor equation always appear in the same (upper or lower) position throughout every term, except for indices repeated in a term (which implies a summation over that index), for example:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }=T^{\alpha }{}_{\beta }{}_{\delta }}$

as for a possible invalid expression:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D_{\alpha }{}_{\beta }{}^{\gamma }E^{\delta }}$

In other words, non-repeated indices must be of the same type in every term of the equation. In the above identity α, β, δ line up throughout and γ occurs twice in one term due to a contraction (correctly once as an upper index and once as a lower index), so it's a valid as an expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is inconsistent.

Brackets and punctuation used once where implied

When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.

Symmetric and antisymmetric parts

Symmetric part of tensor

Round brackets ( ) around some or all indices denotes the symmetrized part of the tensor: the sum of the tensor components with those indices permuted, then divided by the number of permutations.

For two symmetrizing indices, there are two indices to sum over and permute:

${\displaystyle A_{(\alpha \beta )\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }+A_{\beta \alpha \gamma \cdots }\right)}$

as an example,

${\displaystyle A_{(\alpha }B^{\beta }{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }+A_{\gamma }B^{\beta }{}_{\alpha }\right).}$

For three symmetrizing indices, there are three indices to sum over and permute:

${\displaystyle A_{(\alpha \beta \gamma )\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }+A_{\alpha \gamma \beta \delta \cdots }+A_{\gamma \beta \alpha \delta \cdots }+A_{\beta \alpha \gamma \delta \cdots }\right).}$

For p symmetrizing indices – sum over all components with those indices permuted:

${\displaystyle A_{(\alpha _{1}\alpha _{2}\cdots \alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {1}{p!}}\sum _{\mathrm {permute} \,(\alpha _{1},\alpha _{2},\cdots ,\alpha _{p})}A_{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha _{p}\cdots \alpha _{q}}.}$

Antisymmetric part of tensor

Square brackets around some or all indices denotes the antisymmetrized part of the tensor. To antisymmetrize, a total is formed with one term per permutation, components arising from an even permutation of the indices are added, while components arising from an odd permutation of the indices are subtracted, then the total is divided by the number of permutations.

For two antisymmetrizing indices:

${\displaystyle A_{\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }-A_{\beta \alpha \gamma \cdots }\right)}$

For three antisymmetrizing indices:

${\displaystyle A_{\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }-A_{\alpha \gamma \beta \delta \cdots }-A_{\gamma \beta \alpha \delta \cdots }-A_{\beta \alpha \gamma \delta \cdots }\right)}$

as an example,

${\displaystyle F_{}={\dfrac {1}{3!}}\left(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }-\partial _{\gamma }F_{\beta \alpha }-\partial _{\beta }F_{\alpha \gamma }-\partial _{\alpha }F_{\gamma \beta }\right)}$

For p antisymmetrizing indices – sum over the permutations ${\displaystyle \sigma }$ of those indices multiplied by the signature of the permutation, divided by the number of permutations:

${\displaystyle {\begin{aligned}A_{\alpha _{p+1}\cdots \alpha _{q}}&={\dfrac {1}{p!}}\sum _{\sigma }\operatorname {sgn}(\sigma )A_{\sigma (\alpha _{1})\cdots \sigma (\alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}\\&={\dfrac {1}{(n-p)!}}\varepsilon _{\alpha _{1}\dots \alpha _{p}\,\beta _{1}\dots \beta _{n-p}}{\dfrac {1}{p!}}\varepsilon ^{\gamma _{1}\dots \gamma _{p}\,\beta _{1}\dots \beta _{n-p}}A_{\gamma _{1}\dots \gamma _{p}\alpha _{p+1}\cdots \alpha _{q}}\\\end{aligned}}}$

where n is the dimensionality of the underlying vector space.

Symmetry and antisymmetry sum

Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices:

${\displaystyle A_{\alpha \beta \gamma \cdots }=A_{(\alpha \beta )\gamma \cdots }+A_{\gamma \cdots }}$

Differentiation

For compactness, derivatives may be indicated by adding indices after a comma or semicolon.

Partial derivative

To indicate partial differentiation of a tensor field with repect to each coordinate variable ${\displaystyle x^{\gamma }}$, a comma is added before an added index of the coordinate variable. This may repeated (without adding further commas):

${\displaystyle A_{\alpha \beta \cdots ,\gamma }=\partial _{\gamma }A_{\alpha \beta \cdots }={\dfrac {\partial }{\partial x^{\gamma }}}A_{\alpha \beta \cdots }}$

${\displaystyle A_{\alpha \beta \cdots ,\gamma \delta }=\partial _{\delta }\partial _{\gamma }A_{\alpha \beta \cdots }={\dfrac {\partial ^{2}}{\partial x^{\delta }\partial x^{\gamma }}}A_{\alpha \beta \cdots }}$

These components do not transform covariantly.

Covariant derivative

To indicate covariant differentiation with repect to a tangent vector ${\displaystyle v^{\gamma }}$, a semicolon is placed before the index of the tangent vector.

For a covariant tensor: ${\displaystyle A^{\alpha }{}_{;\beta }=A^{\alpha }{}_{,\beta }+\Gamma ^{\alpha }{}_{\gamma \beta }A^{\gamma }}$

For a contravariant tensor: ${\displaystyle A_{\alpha ;\beta }=A_{\alpha ,\beta }-\Gamma ^{\gamma }{}_{\alpha \beta }A_{\gamma }}$

where ${\displaystyle \Gamma ^{\alpha }{}_{\beta \gamma }}$ is a Christoffel symbol of the second kind.

The components of this derivative of a tensor field transform covariantly, and hence form another tensor field.

Components of tensors

Given a tensor field and a basis (of linearly independent vector fields), the coefficients of the tensor field in a basis can be determined by evaluating a suitable combination of the basis and dual basis, and inherits the correct indexing. Some examples are listed below.

Throughout, let e be vector fields that constitute a vector basis at each point (a moving frame).

Tensor product

Given two vectors a and b, their tensor product has the components:

${\displaystyle \mathbf {a} \otimes \mathbf {b} =a_{i}b_{j}\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}}$

Whichever vector comes first has the first index of the tensor, i.e. the above is not the same as

${\displaystyle \mathbf {b} \otimes \mathbf {a} =b_{i}a_{j}\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}}$

By raising and lowering indices similar combinations can be found.

Tensor components

The component of a tensor can be written as a multilinear function of basis vectors:

Covariant tensor: ${\displaystyle T_{ijk\cdots }=T(\mathbf {e} _{i},\mathbf {e} _{j},\mathbf {e} _{k},\cdots )}$

Contravariant tensor: ${\displaystyle T^{ijk\cdots }=T(\mathbf {e} ^{i},\mathbf {e} ^{j},\mathbf {e} ^{k},\cdots )}$

Mixed tensor: ${\displaystyle T^{i}{}_{j}{}^{k\cdots }=T(\mathbf {e} ^{i},\mathbf {e} _{j},\mathbf {e} ^{k},\cdots )}$

and the tensor product of basis vectors gives the tensor.

Covariant tensor: ${\displaystyle {\boldsymbol {\mathsf {T}}}=T_{ijk\cdots }\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}\otimes \mathbf {e} ^{k}\cdots }$

similarly for contravariant and mixed tensors.

This calculation of coefficients also applies for some operations that are not tensorial, for instance:

Christoffel symbols

${\displaystyle \nabla _{i}\mathbf {e} _{j}=\Gamma ^{k}{}_{ij}\mathbf {e} _{k}}$

where ${\displaystyle \nabla _{i}\mathbf {e} _{j}\equiv \nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}\equiv \mathbf {e} _{j;i}}$ is the covariant derivative above, defined by the nabla operator ${\displaystyle \nabla }$ (also called Koszul connection on the tangent vector bundle).

Equivalently,

${\displaystyle \Gamma ^{k}{}_{ij}=\mathbf {e} ^{k}\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}}$

Commutator coefficients

${\displaystyle =\gamma ^{k}{}_{ij}\mathbf {e} _{k}}$

where ${\displaystyle }$ is the Lie bracket as below, equivalently

${\displaystyle \gamma ^{k}{}_{ij}=\mathbf {e} ^{k}}$

Notable tensors

Metric tensor

Covariant metric tensor: ${\displaystyle g_{ij}=g(\mathbf {e} _{i},\mathbf {e} _{j})}$

Contravariant metric tensor: ${\displaystyle g^{ij}=g(\mathbf {e} ^{i},\mathbf {e} ^{j})}$

Riemann curvature tensor

${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=dx^{\rho }(R(\partial _{\mu },\partial _{\nu })\partial _{\sigma })}$

Torsion tensor

${\displaystyle T^{c}{}_{ab}=\Gamma ^{c}{}_{ab}-\Gamma ^{c}{}_{ba}-\gamma ^{c}{}_{ab}}$

which follows from

${\displaystyle T=\nabla _{X}Y-\nabla _{Y}X-}$

where X and Y are vector fields and is the Lie bracket of vector fields.

Levi-Civita tensor

The covariant Levi-Civita tensor in an n-D metric space may be defined as the unique (up to a sign) n-form (completely antisymmetric order-n covariant tensor) that obeys the relation

${\displaystyle \left|\epsilon _{\alpha _{1}\dots \alpha _{n}}g^{\alpha _{1}\beta _{1}}\dots g^{\alpha _{n}\beta _{n}}\epsilon _{\beta _{1}\dots \beta _{n}}\right|=n!}$

The choice of sign defines an orientation in the space.

The contravariant Levi-Civita tensor is an n-vector that may be defined by raising each of the indices of the corresponding covariant tensor:

${\displaystyle \epsilon ^{\alpha _{1}\dots \alpha _{n}}=g^{\alpha _{1}\beta _{1}}\dots g^{\alpha _{n}\beta _{n}}\epsilon _{\beta _{1}\dots \beta _{n}}}$

See also

• Differential form
• Exterior calculus
• Four-gradient
• Penrose graphical notation
• Ricci decomposition

References

1. Ricci, Gregorio; Levi-Civita, Tullio (1900), "Méthodes de calcul différentiel absolu et leurs applications" (PDF), Mathematische Annalen, 54 (1–2), Springer: 125–201, doi:10.1007/BF01454201 {{citation}}: Unknown parameter |month= ignored (help)
2. Schouten, Jan A. (1924). R. Courant (ed.). Der Ricci-Kalkül - Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimmensional differential geometry). Grundlehren der mathematischen Wissenschaften (in German). Vol. 10. Berlin: Springer Verlag.
3. J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. p.85-86, §3.5. ISBN 0-7167-0344-0. {{cite book}}: |page= has extra text (help)CS1 maint: multiple names: authors list (link)
4. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.
5. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
6. Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
7. T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601
8. G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
9. Covariant derivative – Mathworld, Wolfram

Books

• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Lovelock, David (1989) . Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
• J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
• D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
• T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601

• Differential geometry
• Tensors