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- Abstract_index_notation abstract "Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.Let V be a vector space, and V* its dual. Consider, for example, a rank-2 covariant tensor . Then h can be identified with a bilinear form on V. In other words, it is a function of two arguments in V which can be represented as a pair of slots:Abstract index notation is merely a labelling of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical): A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an upper index corresponding to a tensor in V) and one label is covariant (a lower index corresponding to a tensor in V*). Thus, for instance, is the trace of a tensor t = tabc over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or duality pairing) between tensor factors of type V and those of type V*.".
- Abstract_index_notation wikiPageID "1674987".
- Abstract_index_notation wikiPageRevisionID "590289046".
- Abstract_index_notation hasPhotoCollection Abstract_index_notation.
- Abstract_index_notation subject Category:Mathematical_notation.
- Abstract_index_notation subject Category:Tensors.
- Abstract_index_notation type Abstraction100002137.
- Abstract_index_notation type Cognition100023271.
- Abstract_index_notation type Concept105835747.
- Abstract_index_notation type Content105809192.
- Abstract_index_notation type Idea105833840.
- Abstract_index_notation type PsychologicalFeature100023100.
- Abstract_index_notation type Quantity105855125.
- Abstract_index_notation type Tensor105864481.
- Abstract_index_notation type Tensors.
- Abstract_index_notation type Variable105857459.
- Abstract_index_notation comment "Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus.".
- Abstract_index_notation label "Abstract index notation".
- Abstract_index_notation label "Indexnotation von Tensoren".
- Abstract_index_notation label "Notation en indice abstrait".
- Abstract_index_notation label "抽象指标记号".
- Abstract_index_notation sameAs Indexnotation_von_Tensoren.
- Abstract_index_notation sameAs Notation_en_indice_abstrait.
- Abstract_index_notation sameAs m.05mgy_.
- Abstract_index_notation sameAs Q1166872.
- Abstract_index_notation sameAs Q1166872.
- Abstract_index_notation sameAs Abstract_index_notation.
- Abstract_index_notation wasDerivedFrom Abstract_index_notation?oldid=590289046.
- Abstract_index_notation isPrimaryTopicOf Abstract_index_notation.