Introduction.

In this entry we give the definition of a tensor array and establish some related terminology and notation. The theory of tensor arrays incorporates a number of other essential topics: basic tensors, tensor transformations, outer multiplication, contraction, inner multiplication, and generalized transposition. These are fully described in their separate entries.

Valences and the space of tensors arrays.

Point-wise addition and scaling give $\tspace{p,q}$ the structure of a a 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 $\tspace{p,0}$ and $\tspace{0,q}$ are just the maps from, respectively, $I^p$ and $I^q$ to $\kfield$ . It is also customary to identify $\tspace{1,0}$ with $\kfield^I$ , the vector space of list vectors indexed by $I$ , and to identify $\tspace{0,1}$ with dual space $\lp\kfield^I\rp^*$ of linear forms on $\kfield^I$ . Finally, $\tspace{0,0}$ can be identified with $\kfield$ itself. In other words, scalars are tensor arrays of zero valence.

Let $X:I^p\times I^q\rightarrow \kfield$ be 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 $i_1,\ldots, i_p, j_1,\ldots,j_q\in I$ we write $$X^{i_1\ldots i_p}_{j_1\ldots j_q}$$ instead of ⁴$$X(i_1,\ldots,i_p;j_1,\ldots, j_p).$$

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.⁵

Notes.

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.

Footnotes

... short¹

The term tensor has other meanings, c.f. the tensor entry.

...http://planetmath.org/encyclopedia/Field.html ²

In physics and differential geometry, $\kfield$ is typically $\reals$ or $\cnums$ .

... indices³

It is advantageous to allow general indexing sets, because one can indicate the use of multiple frames of reference by employing multiple, disjoint sets of indices.

... of⁴

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

... arrays.⁵

It 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.