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Concept
Einstein notation
Summary
In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.
According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set {1, 2, 3},
is simplified by the convention to:
The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. That is, in this context x2 should be understood as the second component of x rather than the square of x (this can occasionally lead to ambiguity). The upper index position in xi is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see below). Typically, (x1 x2 x3) would be equivalent to the traditional (x y z).
In general relativity, a common convention is that
the Greek alphabet is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are μ, ν, ...),
the Latin alphabet is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are i, j, ...),
In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.
An index that is summed over is a summation index, in this case "i ".
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