Einstein notation sums automatically over an index repeated once up and once down, or twice in Euclidean coordinates.
A dummy index is summed over its full specified range. Its name can be changed consistently without changing the expression, as in . A free index specifies a component and cannot be renamed in only one term of an equation. Relabelling dummy indices is useful for tensor contraction arguments involving a symmetric tensor and an antisymmetric tensor.
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Einstein notation, also known as Einstein summation convention, is a notational scheme used primarily in the fields of mathematics and physics to simplify expressions involving tensors and multi-indexed arrays. It was introduced by the physicist Albert Einstein in the context of his work on the theory of relativity. The key principle of Einstein notation is that when an index variable appears twice in a single term, it implies a summation over that index.
The Wikipedia page of this article is basically a masterclass why Wikipedia is useless for learning technical subjects. They are not even able to teach such a simple subject properly there!
Bibliography:
- www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf gives a definition that does not consider upper and lower indexes, it only counts how many times the indices appearTheir definition of the Laplacian is a bit wrong as only one appears in it, they likely meant to have written instead of , related: