Glossary of tensor theory

This is aglossary of tensor theory.For expositions oftensor theoryfrom different points of view, see:

For some history of the abstract theory see alsomultilinear algebra.

Classical notation[edit]

Ricci calculus: The earliest foundation of tensor theory – tensor index notation.^[1]

Order of a tensor: The components of a tensor with respect to a basis is an indexed array. Theorderof a tensor is the number of indices needed. Some texts may refer to the tensor order using the termdegreeorrank.

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: Adyadictensor is a tensor of order two, and may be represented as a squarematrix.In contrast, adyadis 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, ifa_ijis a matrix, then under this conventiona_iiis itstrace.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

Covarianttensor

Contravarianttensor: The classical interpretation is by components. For example, in the differential forma_idxⁱthecomponentsa_iare 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 ofcontinuum mechanics,such asfluid mechanicsandelasticity.In classicalcontinuum mechanics,the space of interest is usually 3-dimensionalEuclidean space,as is the tangent space at each point. If we restrict the local coordinates to beCartesian coordinateswith the same scale centered at the point of interest, themetric tensoris theKronecker delta.This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors andtensor densities.AllCartesian-tensorindices are written as subscripts.Cartesian tensorsachieve considerable computational simplification at the cost of generality and of some theoretical insight.

Contraction of a tensor

Raising and lowering indices

Symmetric tensor

Antisymmetric tensor

Multiple cross products

Algebraic notation[edit]

This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

Tensor product

Ifvandware vectors invector spacesVandWrespectively, then

v\otimes w

is a tensor in

V\otimes W.

That is, the ⊗ operation is abinary operation,but it takes values into a fresh space (it is in a strong senseexternal). The ⊗ operation is abilinear map;but no other conditions are applied to it.

Pure tensor: A pure tensor ofV⊗Wis one that is of the formv⊗w.; It could be written dyadicallyaⁱb^j,or more accuratelyaⁱb^je_i⊗f_j,where thee_iare a basis forVand thef_ja basis forW.Therefore, unlessVandWhave the same dimension, the array of components need not be square. Suchpuretensors are not generic: if bothVandWhave 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 seeSegre embedding.

Tensor algebra: In the tensor algebraT(V) of a vector spaceV,the operation $\otimes$ becomes a normal (internal)binary operation.A consequence is thatT(V) has infinite dimension unlessVhas dimension 0. Thefree algebraon a setXis for practical purposes the same as the tensor algebra on the vector space withXas basis.

Hodge star operator

Exterior power

Thewedge productis the anti-symmetric form of the ⊗ operation. The quotient space ofT(V) on which it becomes an internal operation is theexterior algebraofV;it is agraded algebra,with the graded piece of weightkbeing called thek-thexterior powerofV.