Editing Glossary of tensor theory (section)

==Classical notation==

;[[Ricci calculus]]
:The earliest foundation of tensor theory – tensor index notation.<ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last1=Ricci|first1=Gregorio|author-link=Gregorio Ricci-Curbastro|last2=Levi-Civita|first2=Tullio|journal=Mathematische Annalen|publisher=Springer|volume=54|issue=1–2|date=March 1900|pages=125–201|doi=10.1007/BF01454201|s2cid=120009332|url=https://zenodo.org/record/1428270|language=fr|trans-title=Absolute differential calculation methods & their applications}}</ref>

;[[Tensor order|Order of a tensor]]
: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''.

;[[Tensor rank|Rank of a tensor]]
: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.

;[[Dyadic tensor]]
:A ''dyadic'' tensor is a tensor of order two, and may be represented as a square [[matrix (mathematics)|matrix]]. In contrast, a ''dyad'' is specifically a dyadic tensor of rank one.

;[[Einstein notation]]
: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 ''a<sub>ij</sub>'' is a matrix, then under this convention ''a<sub>ii</sub>'' is its [[trace (matrix)|trace]]. 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.

;[[Kronecker delta]]

;[[Levi-Civita symbol]]

;[[Covariance and contravariance of vectors|Covariant]] tensor

;[[Covariance and contravariance of vectors|Contravariant]] tensor
:The classical interpretation is by components. For example, in the differential form ''a<sub>i</sub>dx<sup>i</sup>'' the '''components''' ''a<sub>i</sub>'' are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

;[[Mixed tensor]]
:This refers to any tensor that has both lower and upper indices.

;Cartesian tensor
:Cartesian tensors are widely used in various branches of [[continuum mechanics]], such as [[fluid mechanics]] and [[Elasticity (physics)|elasticity]]. In classical [[continuum mechanics]], the space of interest is usually 3-dimensional [[Euclidean space]], as is the tangent space at each point.  If we restrict the local coordinates to be [[Cartesian coordinates]] with the same scale centered at the point of interest, the [[metric tensor]] is the [[Kronecker delta]].  This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and [[Tensor density|tensor densities]].  All [[Cartesian tensor|Cartesian-tensor]] indices are written as subscripts. [[Cartesian tensor]]s achieve considerable computational simplification at the cost of generality and of some theoretical insight.

;[[Tensor contraction|Contraction of a tensor]]

;[[Raising and lowering indices]]

;[[Symmetric tensor]]

;[[Antisymmetric tensor]]

;[[Multiple cross products]]