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- Tensor abstract "Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix (a 2-dimensional array) and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors.Tensors are used to represent correspondences between sets of geometric vectors. For example, the Cauchy stress tensor T takes a direction v as input and produces the stress T(v) on the surface normal to this vector for output thus expressing a relationship between these two vectors, shown in the figure (right).Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. Finding the representation of a tensor in terms of a coordinate basis results in an organized multidimensional array representing the tensor in that basis or frame of reference. The coordinate independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. The precise form of the transformation law determines the type (or valence) of the tensor.Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as elasticity, fluid mechanics, and general relativity. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.".
- Tensor thumbnail Components_stress_tensor.svg?width=300.
- Tensor wikiPageExternalLink 0403252.
- Tensor wikiPageExternalLink books?as_isbn=140201015X.
- Tensor wikiPageExternalLink 2604.html.
- Tensor wikiPageExternalLink 2502.
- Tensor wikiPageExternalLink 3609.
- Tensor wikiPageExternalLink tensors.
- Tensor wikiPageExternalLink Tensors_TM2002211716.pdf.
- Tensor wikiPageExternalLink 978-0-8176-4714-8.
- Tensor wikiPageExternalLink fulltext.pdf.
- Tensor wikiPageID "29965".
- Tensor wikiPageRevisionID "605315635".
- Tensor align "right".
- Tensor caption "Orientation defined by an ordered set of vectors.".
- Tensor caption "Reversed orientation corresponds to negating the exterior product.".
- Tensor footer "Geometric interpretation of grade n elements in a real exterior algebra for , 1 , 2 , 3 . The exterior product of n vectors can be visualized as any n-dimensional shape ; with magnitude , and orientation defined by that on its -dimensional boundary and on which side the interior is.".
- Tensor hasPhotoCollection Tensor.
- Tensor id "3112".
- Tensor image "N vector negative png version.png".
- Tensor image "N vector positive png version.png".
- Tensor title "tensor".
- Tensor width "220".
- Tensor subject Category:Concepts_in_physics.
- Tensor subject Category:Tensors.
- Tensor comment "Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values.".
- Tensor label "Cálculo tensorial".
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- Tensor label "Tensor".
- Tensor label "Tensor".
- Tensor label "Tensor".
- Tensor label "Tensor".
- Tensor label "Tensor".
- Tensor label "Tensore".
- Tensor label "Тензор".
- Tensor label "موتر".
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- Tensor label "張量".
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- Tensor wasDerivedFrom Tensor?oldid=605315635.
- Tensor depiction Components_stress_tensor.svg.
- Tensor isPrimaryTopicOf Tensor.