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Einstein Summation


Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. There are essentially three rules of Einstein summation notation, namely:

1. Repeated indices are implicitly summed over.

2. Each index can appear at most twice in any term.

3. Each term must contain identical non-repeated indices.

The first item on the above list can be employed to greatly simplify and shorten equations involving tensors. For example, using Einstein summation,

 a_ia_i=sum_(i)a_ia_i
(1)

and

 a_(ik)a_(ij)=sum_(i)a_(ik)a_(ij).
(2)

The second and third items on the list indicate that the expression

 M_(ij)v_j=sum_(j)M_(ij)v_j
(3)

is valid, whereas the expressions

 M_(ij)u_jv_j+w_i
(4)

and

 T_(ijk)u_k+M_(ip)
(5)

are invalid because the index j appears three times in the first term of (), while the non-repeated index j in the first term of () doesn't match the non-repeated p of the second term.

The convention was introduced by Einstein (1916, sec. 5), who later jested to a friend, "I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." (Kollros 1956; Pais 1982, p. 216).

In practice, the convention tends to occur alongside both the Kronecker delta and permutation symbol. Moreover, the Einstein summation convention easily accommodates both superscripts and subscripts for contravariant and covariant tensors, respectively.


See also

Contravariant Tensor, Covariant Tensor, Kronecker Delta, Matrix, Permutation Symbol, Sum, Tensor, Tensor Contraction, Vector

Portions of this entry contributed by Christopher Stover

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References

Cubitt, T. "Einstein Summation Convention and delta-Functions." http://www.dr-qubit.org/teaching/summation_delta.pdf.Einstein, A. "Die Grundlage der allgemeinen Relativitätstheorie." Ann. der Physik 49, 769-822, 1916.Kollros, L. "Albert Einstein en Suisse Souvenirs." Helv. Phys. Acta. Supp. 4, 271-281, 1956.Pais, A. Subtle is the Lord: The Science and the Life of Albert Einstein. New York: Oxford University Press, p. 216, 1982.

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Einstein Summation

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Einstein Summation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EinsteinSummation.html

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