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tensor, Tensor, Tensor - Approaches in detail, Tensor - Brief overview, Tensor - Examples, Tensor - History, Tensor - Importance and usage, Tensor - Reference books, Tensor - Tensor densities, Tensor - Tensor software, Tensor - The choice of approach, Tensor - Applications, Tensor - Foundational, Tensor - Notation, Glossary of tensor theory
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ARTICLES RELATED TO tensor | | | |  tensor: Encyclopedia II - Tensor - HistoryThe word "tensor" was first introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus. The word was used in its current meaning by Woldemar Voigt in 1899.
The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry, and was made accessible to many mathematicians by the publication of Tullio Levi-Civita's classic text The Absolute Differential Calculus in 1900 (in Italian; translations followed). The tensor calculus achieved broa ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - History |
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| | | | tensor: Encyclopedia II - Tensor - Examples
A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range. These vectors in the domain are vectors of counting numbers, and these numbers are called indexes. For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the vectors range from <1, 1, 1> through <2, 5, 7>. Here, the tensor would have one value at <1, 1, 1>, another at <1, 1, 2>, and so on for a total of 70 values. (Likewise, vectors may be expressed as a sequence of values ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - Examples |
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| | | | tensor: Encyclopedia II - Tensor product - Tensor product of two tensors What does the general formula mean? It means that if a pair of tensors are juxtaposed (placed side-by-side) then they combine by mere aggregation to form a new tensor which can be subsequently called the tensor product of the pair of juxtaposed tensors. The number of independent components multiplies
There is a general formula for the product of two (or more) tensors, as
.
We are assuming here orthogonal tensors, with no distinction of covariant and contravariant indices, for simplicity.
The parameters introduced above work out like this:
See also:Tensor product, Tensor product - Tensor product of two tensors, Tensor product - Example, Tensor product - Kronecker product of two matrices, Tensor product - Tensor product of multilinear maps, Tensor product - Tensor product of vector spaces, Tensor product - Universal property of tensor product, Tensor product - Tensor product of Hilbert spaces, Tensor product - Definition, Tensor product - Properties, Tensor product - Examples and applications, Tensor product - Relation with the dual space, Tensor product - Types of tensors e.g. alternating, Tensor product - Over more general rings, Tensor product - Tensor product for computer programmers, Tensor product - Array programming languages, Tensor product - C language, Tensor product - SQL Read more here: » Tensor product: Encyclopedia II - Tensor product - Tensor product of two tensors |
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| | | | tensor: Encyclopedia II - Tensor - Importance and usage Tensors are of importance in physics and engineering. In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain. Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general line ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - Importance and usage |
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| | | | tensor: Encyclopedia II - Tensor - Approaches in detail There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.
The classical approach
The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the values in the array. This idea can then be further generalized to tensor fields, w ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - Approaches in detail |
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| | | | tensor: Encyclopedia II - Tensor - Brief overview Depending on what is known as the rank of a tensor, it can be considered to represent any of the more familiar types of quantity as shown below.
* |a| is the determinant of the coefficient array amn or its corresponding in the given dimension. Note that quantities that transform according to column 4 are usually called tensor densities. In case r = 0, the transform in column 4 is just the transform for usual tensors.
It can be deduced from the above that a rank 3 tens ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - Brief overview |
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| | | | tensor: Encyclopedia II - Tensor contraction - Contraction of a tensor with itself Given a mixed tensor of type (m, n) with m≥1 and n≥1, then letting a pair of indices, one contravariant and one covariant, be labeled with the same letter will imply a summation over those two indices. The result of the summation will be a new tensor of type (m−1, n−1) which will inherit the indices of the pre-contracted tensor except for the pair of indices which were bound to each other and over which the contraction took place. Example:
Tαβ< ...
See also:Tensor contraction, Tensor contraction - Contraction of a tensor with itself, Tensor contraction - Contraction of a dyadic tensor, Tensor contraction - Tensor divergence, Tensor contraction - Contraction of a pair of tensors, Tensor contraction - Matrix multiplication, Tensor contraction - Contraction between tensors seen as a self-contraction of a composite tensor Read more here: » Tensor contraction: Encyclopedia II - Tensor contraction - Contraction of a tensor with itself |
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| | | | tensor: Encyclopedia II - Tensor - The choice of approach There are two ways of approaching the definition of tensors:
The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.
The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Covariant vectors, for instance, can also be described as one-forms, or as the elements of th ...
See also:Tensor, Tensor - Brief overview, Tensor - Importance and usage, Tensor - History, Tensor - The choice of approach, Tensor - Examples, Tensor - Approaches in detail, Tensor - Tensor densities, Tensor - Notation, Tensor - Foundational, Tensor - Applications, Tensor - Reference books, Tensor - Tensor software Read more here: » Tensor: Encyclopedia II - Tensor - The choice of approach |
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| | | | tensor: Encyclopedia II - Tensor field - Notation The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as
to emphasize that the tangent bundle is the range space of the (1,0) tensor fields on the manifold M. Do not confuse this with the very similar looking notation
;
in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined ...
See also:Tensor field, Tensor field - Geometric introduction, Tensor field - The vector bundle explanation, Tensor field - Notation, Tensor field - The C∞M module explanation, Tensor field - Applications, Tensor field - Tensor calculus, Tensor field - Twisting by a line bundle, Tensor field - The flat case, Tensor field - Cocycles and chain rules Read more here: » Tensor field: Encyclopedia II - Tensor field - Notation |
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| | | | tensor: Encyclopedia II - Tensor product - Tensor product for computer programmers
Tensor product - Array programming languages.
Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as (for example or ). In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c).
Note that J's treatment also allows the representation of some tensor fields (as a and b may be functions instead of constants -- the result is then a derived function, and if a ...
See also:Tensor product, Tensor product - Tensor product of two tensors, Tensor product - Example, Tensor product - Kronecker product of two matrices, Tensor product - Tensor product of multilinear maps, Tensor product - Tensor product of vector spaces, Tensor product - Universal property of tensor product, Tensor product - Tensor product of Hilbert spaces, Tensor product - Definition, Tensor product - Properties, Tensor product - Examples and applications, Tensor product - Relation with the dual space, Tensor product - Types of tensors e.g. alternating, Tensor product - Over more general rings, Tensor product - Tensor product for computer programmers, Tensor product - Array programming languages, Tensor product - C language, Tensor product - SQL Read more here: » Tensor product: Encyclopedia II - Tensor product - Tensor product for computer programmers |
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| | | | tensor: Encyclopedia II - Tensor product - Tensor product of vector spaces The tensor product of two vector spaces V and W has a formal definition by the method of generators and relations. The equivalence class under these relations (given below) of (v,w) is called a tensor and is denoted by . By construction, one can prove several identities between tensors and form an algebra of tensors.
To construct , take the vector space generated by and apply (factor out the subspace generated by) the followi ...
See also:Tensor product, Tensor product - Tensor product of two tensors, Tensor product - Example, Tensor product - Kronecker product of two matrices, Tensor product - Tensor product of multilinear maps, Tensor product - Tensor product of vector spaces, Tensor product - Universal property of tensor product, Tensor product - Tensor product of Hilbert spaces, Tensor product - Definition, Tensor product - Properties, Tensor product - Examples and applications, Tensor product - Relation with the dual space, Tensor product - Types of tensors e.g. alternating, Tensor product - Over more general rings, Tensor product - Tensor product for computer programmers, Tensor product - Array programming languages, Tensor product - C language, Tensor product - SQL Read more here: » Tensor product: Encyclopedia II - Tensor product - Tensor product of vector spaces |
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| | | | tensor: Encyclopedia II - Tensor product - Universal property of tensor product The space of all bilinear maps from to is naturally isomorphic to the space of all linear maps from to . This is built into the construction; has all and only the relations that are necessary to ensure that a homomorphism from to will be linear.
The tensor product in fact satisfies the universal property of being a fibered coproduct.
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See also:Tensor product, Tensor product - Tensor product of two tensors, Tensor product - Example, Tensor product - Kronecker product of two matrices, Tensor product - Tensor product of multilinear maps, Tensor product - Tensor product of vector spaces, Tensor product - Universal property of tensor product, Tensor product - Tensor product of Hilbert spaces, Tensor product - Definition, Tensor product - Properties, Tensor product - Examples and applications, Tensor product - Relation with the dual space, Tensor product - Types of tensors e.g. alternating, Tensor product - Over more general rings, Tensor product - Tensor product for computer programmers, Tensor product - Array programming languages, Tensor product - C language, Tensor product - SQL Read more here: » Tensor product: Encyclopedia II - Tensor product - Universal property of tensor product |
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| | | | tensor: Encyclopedia II - Tensor product - Tensor product of Hilbert spaces The tensor product of two Hilbert spaces is another Hilbert space, which is defined as described below.
Tensor product - Definition.
Let H1 and H2 be two Hilbert spaces with inner products and , respectively. Construct the tensor product of H1 and H2 as vector spaces as explained above. We can turn this vector space tensor p ...
See also:Tensor product, Tensor product - Tensor product of two tensors, Tensor product - Example, Tensor product - Kronecker product of two matrices, Tensor product - Tensor product of multilinear maps, Tensor product - Tensor product of vector spaces, Tensor product - Universal property of tensor product, Tensor product - Tensor product of Hilbert spaces, Tensor product - Definition, Tensor product - Properties, Tensor product - Examples and applications, Tensor product - Relation with the dual space, Tensor product - Types of tensors e.g. alternating, Tensor product - Over more general rings, Tensor product - Tensor product for computer programmers, Tensor product - Array programming languages, Tensor product - C language, Tensor product - SQL Read more here: » Tensor product: Encyclopedia II - Tensor product - Tensor product of Hilbert spaces |
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