Glossary of tensor theory
This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
For some history of the abstract theory see also Multilinear algebra.
Classical notation
The earliest foundation of tensor theory - tensor index notation.[1]
A tensor written in component form is an indexed array. The order of a tensor is the number of indices required. (The rank of tensor used to mean the order, but now it means something different)
- Rank
The rank of the tensor is the minimal number of rank-one tensor that you need to sum up to obtain this higher-rank tensor. Rank-one tensors are given the generalization of outer product to m-vectors where m is the order of the tensor.
A dyadic tensor has order two, and may be represented as a square matrix. The conventions aij , aij , and aij , do have different meanings (the position of the index determines its valence (variance), in that the first may represent a quadratic form, the second a linear transformation, and the distinction is important in contexts that require tensors that aren't orthogonal (see below). A dyad is a tensor such as aibj, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense of linear algebra - a clashing terminology that can cause confusion.
This notation is based on the understanding that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index. For example if aij is a matrix, then under this convention aii is its trace. The Einstein convention is generally used in physics and engineering texts, to the extent that if summation is not to be applied it is normal to note that explicitly.
- Covariant tensor
- Contravariant 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.
This refers to any tensor with lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and 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 densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.
Algebraic notation
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 vector spaces V and W respectively, then
is a tensor in
That is, the ⊗ operation is a binary operation, but it takes values into a fresh space (it is in a strong sense external). The ⊗ operation is a bilinear map; 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 Segre embedding.
Tensor algebra
In the tensor algebra T(V) of a vector space V, the operation
becomes a normal (internal) binary operation. This is at the cost of T(V) being of infinite dimension, unless V has dimension 0. The free algebra 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 wedge product is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, 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 polynomial algebras.
Applications
Tensor field theory
Abstract algebra
This is an operation on fields, that does not always produce a field.
A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.
These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comes from the torsion subgroup in abelian group theory.
These are highly abstract approaches used in some parts of geometry.
Spinors
See:
References
- ^ Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" (PDF), Mathematische Annalen, 54 (1–2), Springer: 125–201, doi:10.1007/BF01454201
Books
- Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
- Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
- Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
- Lovelock, David (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
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