Glossary of tensor theory

<h2 id="classical-notation">Classical notation</h2>
<a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a>
The earliest foundation of tensor theory – tensor index notation.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
<a href="/facts/Tensor_order/pNjFo5tV">Order of a tensor</a>
The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank.
<a href="/facts/Tensor_rank/2KuNm0MK">Rank of a tensor</a>
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.
<a href="/facts/Dyadic_tensor/FoLJnrwR">Dyadic tensor</a>
A dyadic tensor is a tensor of order two, and may be represented as a square <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>. In contrast, a dyad is specifically a dyadic tensor of rank one.
<a href="/facts/Einstein_notation/UXwS43wU">Einstein notation</a>
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 aij is a matrix, then under this convention aii is its <a href="/facts/Trace_(matrix)/dchFFvHt">trace</a>. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.
<a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>
<a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>
<a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">Covariant</a> tensor
<a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">Contravariant</a> tensor
The classical interpretation is by components. For example, in the differential form aidxi the components ai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.
<a href="/facts/Mixed_tensor/I5H8r3a2">Mixed tensor</a>
This refers to any tensor that has both lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches of <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, such as <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a> and <a href="/facts/Elasticity_(physics)/wcJoIFg1">elasticity</a>. In classical <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, the space of interest is usually 3-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, as is the tangent space at each point. If we restrict the local coordinates to be <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> with the same scale centered at the point of interest, the <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and <a href="/facts/Tensor_density/XF4gqBCM">tensor densities</a>. All <a href="/facts/Cartesian_tensor/L1fJoNnY">Cartesian-tensor</a> indices are written as subscripts. <a href="/facts/Cartesian_tensor/L1fJoNnY">Cartesian tensors</a> achieve considerable computational simplification at the cost of generality and of some theoretical insight.
<a href="/facts/Tensor_contraction/fqz0zvWp">Contraction of a tensor</a>
<a href="/facts/Raising_and_lowering_indices/cPsSVpgI">Raising and lowering indices</a>
<a href="/facts/Symmetric_tensor/CWKBfgnv">Symmetric tensor</a>
<a href="/facts/Antisymmetric_tensor/atcSaPST">Antisymmetric tensor</a>
<a href="/facts/Multiple_cross_products/46u8v6SV">Multiple cross products</a>
<h2 id="algebraic-notation">Algebraic notation</h2>
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

Tensor product
If v and w are vectors in <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> V and W respectively, then

v
        ⊗
        w
      
    
    {\displaystyle v\otimes w}

is a tensor in

V
        ⊗
        W
        .
      
    
    {\displaystyle V\otimes W.}

That is, the ⊗ operation is a <a href="/facts/Binary_operation/CYtow0bz">binary operation</a>, but it takes values into a fresh space (it is in a strong sense external). The ⊗ operation is a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a>; but no other conditions are applied to it.
Pure tensor
A pure tensor of V ⊗ W is one that is of the form v ⊗ w.
It could be written dyadically aibj, or more accurately aibj ei ⊗ fj, where the ei are a basis for V and the fj a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square. Such pure tensors are not generic: if both V and W have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see <a href="/facts/Segre_embedding/xDVUkB2C">Segre embedding</a>.
Tensor algebra
In the tensor algebra T(V) of a vector space V, the operation 
 
 
 
 ⊗
 
 
 {\displaystyle \otimes }
 
 becomes a normal (internal) <a href="/facts/Binary_operation/CYtow0bz">binary operation</a>. A consequence is that T(V) has infinite dimension unless V has dimension 0. The <a href="/facts/Free_algebra/xMrT0WbT">free algebra</a> on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
Hodge star operator
Exterior power
The <a href="/facts/Wedge_product/rdN4TvpR">wedge product</a> is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomes an internal operation is the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> of V; it is a <a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a>, with the graded piece of weight k being called the k-th exterior power of V.
Symmetric power, symmetric algebra
This is the invariant way of constructing <a href="/facts/Polynomial_algebra/ua3vk8Ih">polynomial algebras</a>.
<h2 id="applications">Applications</h2>
<a href="/facts/Metric_tensor/tbqek1gJ">Metric tensor</a>
<a href="/facts/Strain_tensor/j4CJFdsU">Strain tensor</a>
<a href="/facts/Stress%E2%80%93energy_tensor/wWKrzjfI">Stress–energy tensor</a>
<h2 id="tensor-field-theory">Tensor field theory</h2>
<a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a>
<a href="/facts/Tensor_field/cAC6F6rF">Tensor field</a>
<a href="/facts/Tensor_density/XF4gqBCM">Tensor density</a>
<a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a>
<a href="/facts/Tensor_derivative/sRGN0Xh6">Tensor derivative</a>
<a href="/facts/Differential_geometry/c04XmyLu">Differential geometry</a>
<h2 id="abstract-algebra">Abstract algebra</h2>
<a href="/facts/Tensor_product_of_fields/FklJQo5j">Tensor product of fields</a>
This is an operation on fields, that does not always produce a field.
<a href="/facts/Tensor_product_of_R-algebras/TdWszpzf">Tensor product of R-algebras</a>
<a href="/facts/Clifford_module/VgEkUgDT">Clifford module</a>
A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.
<a href="/facts/Tor_functor/oFn2L3gd">Tor functors</a>
These are the <a href="/facts/Derived_functor/YdjtF5ds">derived functors</a> of the tensor product, and feature strongly in <a href="/facts/Homological_algebra/Jfm8w52l">homological algebra</a>. The name comes from the <a href="/facts/Torsion_subgroup/dIwLuG5j">torsion subgroup</a> in <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> theory.
<a href="/facts/Symbolic_method_of_invariant_theory/LwWxKfUq">Symbolic method of invariant theory</a>
<a href="/facts/Derived_category/pBQHnFYo">Derived category</a>
<a href="/facts/Grothendieck%27s_six_operations/lAz8n9tc">Grothendieck's six operations</a>
These are highly abstract approaches used in some parts of geometry.
<h2 id="spinors">Spinors</h2>
See:

<a href="/facts/Spin_group/tfvR9cqM">Spin group</a>
<a href="/facts/Spin-c_group/BF78kFz8">Spin-c group</a>
<a href="/facts/Spinor/sKFNnEwP">Spinor</a>
<a href="/facts/Pin_group/1lHT68gC">Pin group</a>
<a href="/facts/Pinor/1lHT68gC">Pinors</a>
<a href="/facts/Spinor_field/z88hf7ps">Spinor field</a>
<a href="/facts/Killing_spinor/qFS8KUNN">Killing spinor</a>
<a href="/facts/Spin_manifold/BF78kFz8">Spin manifold</a>

<h2 id="books">Books</h2>
<ul><li><a href="/facts/Richard_L._Bishop/pvUt9kNG">Bishop, R.L.</a>; Goldberg, S.I. (1968), <a href="https://archive.org/details/tensoranalysison00bish">Tensor Analysis on Manifolds</a> (First Dover 1980 ed.), The Macmillan Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-64039-6</li>
<li><a href="/facts/Donald_A._Danielson/nsuH9jN3">Danielson, Donald A.</a> (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8133-4080-7.</li>
<li>Dimitrienko, Yuriy (2002). <a href="https://books.google.com/books?as_isbn=140201015X">Tensor Analysis and Nonlinear Tensor Functions</a>. Kluwer Academic Publishers (Springer). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-1015-X.</li>
<li>Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-65840-7.</li>
<li><a href="/facts/John_L._Synge/p2Qtx5um">Synge, John L</a>; <a href="/facts/Alfred_Schild/oXpNIwN8">Schild, Alfred</a> (1949). <a href="https://archive.org/details/tensorcalculus00syng">Tensor Calculus</a>. <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a> 1978 edition. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-63612-2.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" [Absolute differential calculation methods & their applications], Mathematische Annalen (in French), 54 (1–2), Springer: 125–201, doi:10.1007/BF01454201, S2CID 120009332 <a href="/wiki/Gregorio_Ricci-Curbastro" target="_blank">/wiki/Gregorio_Ricci-Curbastro</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Glossary of tensor theory open-in-new

Glossary of tensor theory