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,

(1)

and

(2)

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

(3)

is valid, whereas the expressions

(4)

and

(5)

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

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