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  2. Index notation - Wikipedia

    en.wikipedia.org/wiki/Index_notation

    Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]

  3. Abstract index notation - Wikipedia

    en.wikipedia.org/wiki/Abstract_index_notation

    Abstract index notation (also referred to as slot-naming index notation) [1] is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. [2] The indices are mere placeholders, not related to any basis and, in particular, are non-numerical.

  4. Raising and lowering indices - Wikipedia

    en.wikipedia.org/wiki/Raising_and_lowering_indices

    Concretely, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K {\displaystyle \alpha :V\rightarrow K} .

  5. Multi-index notation - Wikipedia

    en.wikipedia.org/wiki/Multi-index_notation

    Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

  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. Tensor - Wikipedia

    en.wikipedia.org/wiki/Tensor

    The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation.

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  9. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    Replacing any index symbol throughout by another leaves the tensor equation unchanged (provided there is no conflict with other symbols already used). This can be useful when manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-Civita symbol (see also below). An ...