We'll end this discussion of tensors in physics with a story. I was the math consultant for the 4th edition of the American Heritage Dictionary of the English Language (2000). The editors sent me all the words in the 3rd edition with mathematical definitions, and I had to correct any errors. Early on I came across a word I had never heard of before: dyad. It was defined in the 3rd edition as "an operator represented as a pair of vectors juxtaposed without multiplication." That's a ridiculous definition, as it conveys no meaning at all. I obviously had to fix this definition, but first I had to know what the word meant! In a physics book27 a dyad is defined as "a pair of vectors, written in a definite order ab." This is just as useless, but the physics book also does something with dyads, which gives a clue about what they really are. The product of a dyad ab with a vector c is a(b • c), where b • c is the usual dot product (a, b, and c are all vectors in Rn). This reveals what a dyad is. Do you see it? Dotting with b is an element of the dual space ( Rn)V, so the effect of ab on c is reminiscient of the way \( V \otimes V^V\)
acts on V by
\( (v \otimes \varphi)(w) = \varphi(w)v \)
. A dyad is the same thing as an elementary tensor
\( v \otimes \varphi \)
in
\( \mathbf{R}^n \otimes (\mathbf{R}^n)^V \)
. In the 4th edition of the dictionary, I included two definitions for a dyad. For the general reader, a dyad is "a function that draws a correspondence
28 from any vector
u to the vector (
v •
u)
w and is denoted
vw, where
v and
w are a fixed pair of vectors and
v •
u is the scalar product of
v and
u. For example, if
v = (2, 3, 1),
w = (0, -1, 4), and
u = (a,b,c), then the dyad
vw draws a correspondence from
u to (2 a + 3 b + c )
w ." The more concise second definition was: a dyad is "a tensor formed from a vector in a vector space and a linear functional on that vector space." Unfortunately, the definition of "tensor" in the dictionary is "A set of quantities that obey certain transformation laws relating the bases in one generalized coordinate system to those of another and involving partial derivative sums. Vectors are simple tensors." That is really the definition of a tensor field, and that sense of the word tensor is incompatible with my concise definition of a dyad in terms of tensors.
More general than a dyad is a
dyadic, which is a sum of dyads:
ab +
cd + .... So a dyadic is a general tensor in
\( \mathbf{R}^n \otimes_{\mathbf{R}} (\mathbf{R}^n)^V \cong \mbox{Hom}_{\mathbf{R}} ( \mathbf{R}^n , \mathbf{R}^n) \)
. In other words, a dyadic is an n × n real matrix. The terminology of dyads and dyadics goes back to Gibbs [4, Chap. 3], who championed the development of linear and multilinear algebra, including his indeterminate product (that is, the tensor product), under the name "multiple algebra."
27 H. Goldstein,
Classical Mechanics , 2nd ed., p. 194
28 Yes, this terminology sucks. Blame the unknown editor at the dictionary for that one.[/size]
)
