tensor array (original) (raw)

Introduction.

, or tensors for short11The term tensor has other meanings, c.f. the tensor entry (http://planetmath.org/Tensor).are multidimensional arrays with two types of (covariant and contravariant) indices. Tensors are widely used in science and mathematics, because these data structures are the natural choice ofrepresentation for a variety of important physical and geometric quantities.

Valences and the space of tensors arrays.

Let 𝕂 be a field22In physics and differential geometry, 𝕂 is typically ℝ or ℂ. and let Ibe a finite list of indices33It is advantageous to allow general indexing sets, because one can indicate the use of multipleframes of reference by employing multiple, disjoint sets of indices., such as (1,2,…,n). A tensor array of type

is a mapping

The set of all such mappings will be denoted by Tp,q⁢(I,𝕂), or when I and𝕂 are clear from the context, simply as Tp,q. The numbers p and q are called, respectively, the contravariant and the covariant valence of the tensor array.

Point-wise addition and scaling give Tp,q the structure of a a vector space of dimension np+q, where n is the cardinality ofI. We will interpret I0 as signifying a singleton set. Consequently Tp,0 and T0,q are just the maps from, respectively, Ip and Iq to 𝕂. It is also customary to identify T1,0 with 𝕂I, the vector space of list vectors indexed by I, and to identify T0,1 with dual space (𝕂I)* of linear forms on 𝕂I. Finally,T0,0 can be identified with 𝕂 itself. In other words, scalars are tensor arrays of zero valence.

LetX:Ip×Iq→𝕂be a type (p,q) tensor array. In writing the values of X, it is customary to write contravariant indices using superscripts, and covariant indices using subscripts. Thus, for indices i1,…,ip,j1,…,jq∈I we write

instead of 44Curiously, the latter notation is preferred by some authors. See H. Weyl’s books and papers, for example.

We also mention that it is customary to use columns to represent contravariant index dimensions, and rows to represent the covariant index dimensions. Thus column vectors are type (1,0) tensor arrays,row vectors are type (0,1) tensor arrays, and matrices, in as much as they can be regarded either as rows of columns or as columns of rows, are type (1,1) tensor arrays.55It is also customary to use matrices to also represent type (2,0) and type (0,2) tensor arrays (The latter are used to represent quadratic forms.) Speaking idealistically, such objects should be typeset, respectively, as a column of column vectors and as a row of row vectors. However typographical constraints and notational convenience dictate that they be displayed as matrices.

Notes.

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

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