Editing Tensor (section)

== Examples ==
{{See also|Dyadic tensor}}
<!--NOTE: "Dyadic" is old terminology for matrix representations of rank-2 tensors which could be any of {{math|(2, 0)}}, {{math|(1, 1)}} or {{math|(0, 2)}} (which clashes with bilinear forms, inner products, metric etc.)&nbsp;– its meaning is vague, unclear and confusing, and they aren't really used anymore.  Please do not add "dyadic tensor" to the table.  Thanks. -->

An elementary example of a mapping describable as a tensor is the [[dot product]], which maps two vectors to a scalar. A more complex example is the [[Cauchy stress tensor]] '''T''', which takes a directional unit vector '''v''' as input and maps it to the stress vector '''T'''<sup>('''v''')</sup>, which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to '''v''' against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The [[cross product]], where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The [[Levi-Civita symbol|totally anti-symmetric symbol]] <math>\varepsilon_{ijk}</math> nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.

This table shows important examples of tensors on vector spaces and tensor fields on manifolds.  The tensors are classified according to their type {{math|(''n'', ''m'')}}, where ''n'' is the number of contravariant indices, ''m'' is the number of covariant indices, and {{math|''n'' + ''m''}} gives the total order of the tensor.  For example, a [[bilinear form]] is the same thing as a {{math|(0, 2)}}-tensor; an [[inner product]] is an example of a {{math|(0, 2)}}-tensor, but not all {{math|(0, 2)}}-tensors are inner products.  In the {{math|(0, ''M'')}}-entry of the table, ''M'' denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.

{| class="wikitable"
|+ Example tensors on vector spaces and tensor fields on manifolds
|-
! colspan=2 rowspan=2 width="75px"  |
! colspan=7                         | ''m''
|-
! scope="col" width="175px" | 0
! scope="col" width="175px" | 1
! scope="col" width="175px" | 2
! scope="col" width="175px" | 3
! scope="col" width="75px"  | ⋯
! scope="col" width="175px" | ''M''
! scope="col" width="75px"  | ⋯
|-
! rowspan=6 | ''n''
! scope="row" | 0
| [[Scalar (mathematics)|Scalar]], e.g. [[scalar curvature]]
| [[Covector]], [[linear functional]], [[1-form]], e.g. [[multipole expansion|dipole moment]], [[gradient]] of a scalar field
| [[Bilinear form]], e.g. [[inner product]], [[quadrupole moment]], [[metric tensor]], [[Ricci curvature]], [[2-form]], [[symplectic form]]
| 3-form E.g. [[multipole moment|octupole moment]]
|
| E.g. ''M''-form i.e. [[volume form]]
|
|-
! scope="row" | 1
| [[Euclidean vector]]
| [[Linear transformation]],<ref name="BambergSternberg1991">{{cite book|first1=Paul|last1=Bamberg|first2=Shlomo|last2=Sternberg|title=A Course in Mathematics for Students of Physics|volume=2|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40650-5|page=669}}</ref>  [[Kronecker delta]]
| E.g. [[cross product]] in three dimensions
| E.g. [[Riemann curvature tensor]]
|
|
|
|-
! scope="row" | 2
| Inverse [[metric tensor]], [[bivector]], e.g., [[Poisson structure]]
|
| E.g. [[elasticity tensor]]
|
|
|
|
|-
! scope="row" | ⋮
|
|
|
|
|
|
|
|-
! scope="row" | ''N''
|[[Multivector]]
|
|
|
|
|
|
|-
! scope="row" | ⋮
|
|
|
|
|
|
|
|}

Raising an index on an {{math|(''n'', ''m'')}}-tensor produces an {{math|(''n'' + 1, ''m'' − 1)}}-tensor; this corresponds to moving diagonally down and to the left on the table.  Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table.  [[#Contraction|Contraction]] of an upper with a lower index of an {{math|(''n'', ''m'')}}-tensor produces an {{math|(''n'' − 1, ''m'' − 1)}}-tensor; this corresponds to moving diagonally up and to the left on the table.

{{Clear}}
{{multiple image
   | align = right
   | footer    = Geometric interpretation of grade ''n'' elements in a real [[exterior algebra]] for {{math|1=''n'' = 0}} (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume).  The exterior product of ''n'' vectors can be visualized as any ''n''-dimensional shape (e.g. ''n''-[[Parallelepiped#Parallelotope|parallelotope]], ''n''-[[ellipsoid]]); with magnitude ([[hypervolume]]), and [[Orientation (vector space)|orientation]] defined by that on its {{math|''n'' − 1}}-dimensional boundary and on which side the interior is.<ref>{{cite book |first=R. |last=Penrose| title=The Road to Reality| publisher= Vintage  | year=2007 | isbn=978-0-679-77631-4|title-link=The Road to Reality}}</ref><ref>{{cite book|title=Gravitation|first1=J.A. |last1=Wheeler |first2=C. |last2=Misner |first3=K.S. |last3=Thorne |publisher=W.H. Freeman |year=1973|page=83|isbn=978-0-7167-0344-0|url={{google books |plainurl=y |id=w4Gigq3tY1kC}}}}</ref>
   | width1    = 220
   | image1    = N vector positive.svg
   | caption1  = Orientation defined by an ordered set of vectors.
   | width2    = 220
   | image2    = N vector negative.svg
   | caption2  = Reversed orientation corresponds to negating the exterior product.
}}