tensor array

Introduction.

, or tensors for shortare multidimensional arrays with two types of (covariant andcontravariant) indices. Tensors are widely used in science andmathematics, because these data structures are the natural choice ofrepresentation for a variety of important physical and geometricquantities.

In this entry we give the definition of a tensor array and establishsome related terminology and notation. The theory of tensor arraysincorporates a number of other essential topics: basic tensors, tensortransformations, outer multiplication, contraction, innermultiplication, and generalized transposition. These are fullydescribed in their separate entries.

Valences and the space of tensors arrays.

Let $𝕂$ be a field and let $I$be a finite list of indices, such as $(1,2,\mathrm{\dots },n)$. A tensor array of type

$$(p,q),p,q\in \mathbb{N}$$

is a mapping

$$I^p\times I^q\to 𝕂.$$

The set of all such mappingswill be denoted by ${\mathrm{T}}^{p,q}(I,𝕂)$, or when $I$ and$𝕂$ are clear from the context, simply as ${\mathrm{T}}^{p,q}$. Thenumbers $p$ and $q$ are called, respectively, the contravariant andthe covariant valence of the tensor array.

Point-wise addition and scaling give ${\mathrm{T}}^{p,q}$ the structure of aa vector space of dimension $n^{p+q}$, where $n$ is the cardinality of$I$. We will interpret $I^0$ as signifying a singleton set.Consequently ${\mathrm{T}}^{p,0}$ and ${\mathrm{T}}^{0,q}$ are just the maps from,respectively, $I^p$ and $I^q$ to $𝕂$. It is also customary toidentify ${\mathrm{T}}^{1,0}$ with ${𝕂}^I$, the vector space of listvectors indexed by $I$, and to identify ${\mathrm{T}}^{0,1}$ with dual space${\left({𝕂}^I\right)}^{*}$ of linear forms on ${𝕂}^I$. Finally,${\mathrm{T}}^{0,0}$ can be identified with $𝕂$ itself. In otherwords, scalars are tensor arrays of zero valence.

Let$X:I^p\times I^q\to 𝕂$be a type $(p,q)$ tensor array.In writing the values of $X$, it is customary to write contravariantindices using superscripts, and covariant indices using subscripts.Thus, for indices $i_1,\mathrm{\dots },i_p,j_1,\mathrm{\dots },j_q\in I$ we write

$$X_{j_1\mathrm{\dots }{j}_q}^{i_1\mathrm{\dots }{i}_p}$$

instead of

$$X(i_1,\mathrm{\dots },i_p;j_1,\mathrm{\dots },j_p).$$

We also mention that it is customary to use columns to representcontravariant index dimensions, and rows to represent the covariantindex dimensions. Thus column vectors are type $(1,0)$ tensor arrays,row vectors are type $(0,1)$ tensor arrays, and matrices, in as muchas they can be regarded either as rows of columns or as columns ofrows, are type $(1,1)$ tensor arrays.

Notes.

It must be noted that our usage of the term tensor array isnon-standard. The traditionally inclined authors simply call thesedata structures tensors. We bother to make the distinction becausethe traditional nomenclature is ambiguous and doesn’t include themodern mathematical understanding of the tensor concept. (This isexplained more fully in the tensor entry (http://planetmath.org/Tensor).) Precise andmeaningful definitions can only be given by treating the concept of atensor array as distinct from the concept of a geometric/abstracttensor.

We also mention that the term tensor is often applied to objectsthat should more appropriately be termed a tensor field. Thelatter are tensor-valued functions, or more generally sections of atensor bundle. A tensor is what one gets by evaluating a tensor fieldat one point. Informally, one can also think of a tensor field as atensor whose values are functions, rather than constants.

Title	tensor array
Canonical name	TensorArray
Date of creation	2013-03-22 12:40:25
Last modified on	2013-03-22 12:40:25
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	10
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A69
Related topic	Frame
Related topic	Vector2
Related topic	BasicTensor
Related topic	TensorProductClassical
Related topic	Tensor
Defines	covariant index
Defines	contravariant index
Defines	valence