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  2. Einstein notation - Wikipedia

    en.wikipedia.org/wiki/Einstein_notation

    Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. [2]

  3. Raising and lowering indices - Wikipedia

    en.wikipedia.org/wiki/Raising_and_lowering_indices

    We need not raise or lower all indices at once: it is perfectly fine to raise or lower a single index. Lowering an index of an ( r , s ) {\displaystyle (r,s)} tensor gives a ( r − 1 , s + 1 ) {\displaystyle (r-1,s+1)} tensor, while raising an index gives a ( r + 1 , s − 1 ) {\displaystyle (r+1,s-1)} (where r , s {\displaystyle r,s} have ...

  4. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    Each index has one possible value per dimension of the underlying vector space. The number of indices equals the degree (or order) of the tensor. For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices ...

  5. Levi-Civita symbol - Wikipedia

    en.wikipedia.org/wiki/Levi-Civita_symbol

    In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.

  6. Glossary of tensor theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_tensor_theory

    Einstein notation This notation is based on the understanding that whenever a multidimensional array contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example, if a ij is a matrix, then under this convention a ii is its trace. The Einstein convention is widely ...

  7. Covariance and contravariance of vectors - Wikipedia

    en.wikipedia.org/wiki/Covariance_and_contra...

    In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...

  8. Christoffel symbols - Wikipedia

    en.wikipedia.org/wiki/Christoffel_symbols

    where (g jk) is the inverse of the matrix (g jk), defined as (using the Kronecker delta, and Einstein notation for summation) g ji g ik = δ j k. Although the Christoffel symbols are written in the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates.

  9. Tensor contraction - Wikipedia

    en.wikipedia.org/wiki/Tensor_contraction

    In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual.In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression.