Einstein notation

Einstein notation is a convention in which a repeated index appearing once as an upper index and once as a lower index implies summation over that index.

Einstein summation convention: A notation rule that replaces an explicit sum with a repeated-index expression. It is the summation rule at the center of Einstein notation.
Index notation: A way of expressing mathematical objects and operations using indexed components. Einstein notation is a compact form of index notation.
Tensor calculus: The study of tensor operations and their behavior under coordinate transformations. Einstein notation is a standard shorthand for tensor-calculus equations.
Explicit summation notation: Notation that writes a summation symbol and its index bounds directly. It states sums that Einstein notation leaves implicit.
Tensor contraction: An operation that sums over a matched pair of tensor indices, reducing tensor rank by two. Repeated upper and lower indices encode this operation.
Tensor: A multilinear object whose components transform consistently under changes of coordinates. Tensor components are the quantities Einstein notation most often represents.
General relativity: Einstein’s theory of gravity, in which gravitation is described through spacetime geometry. Its field equations use compact tensor expressions with repeated indices.
Einstein summation convention with unrestricted repeated indices: A generalized convention that sums repeated indices even when both appear in the same index position. It differs from the upper-lower pairing specified by the standard tensor convention.
Kronecker delta: A two-index symbol equal to one when its indices match and zero otherwise. It acts as an identity tensor and can select or rename summed indices.
Covariant and contravariant indices: Lower and upper index types that transform differently under coordinate changes. Their pairing determines which repeated indices imply summation.

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