Diadics and tensors. The motivation for diadics. Nonionic form. Reddy's "Continuum Mechanics."

Question

I'm taking a course in continuum mechanics. Our book is Continuum Mechanics by Reddy, a Cambridge edition. In the second chapter he introduces tensors and defines them to be polyadics. He is specifically concerned with dyadic which are tensors of rank (2,0).

In his presentation of dyadic, the notation is introduced, then some properties are given, but no rigorous definition is present. Then the nonionic form is discussed. This form is a $3$x$3$ matrix, as we work in $\mathbb R^3$. The dyadic can apparently be expressed in "nonionic" form, but the laws for getting the matrix entries are not given, and no sufficient information is available in the book to deduce them. Is there a law governing transition from one form to the other?

Furthermore, is this way of introducing tensors standard? Is it just in this one book? What is the motivation for talking about dyadic?

Answer 1

Dyadics and polyadics are a fantastically archaic way, due to Gibbs of thermodynamics fame, to denote and work with linear transformations and tensors in general. In general, the $n$-adic product of vectors $v_1,\dotsc,v_n \in \mathbb{R}^3$ is simply ${\bf v}_1 \cdots {\bf v}_n := v_1 \otimes \cdots \otimes v_n \in (\mathbb{R}^3)^{\otimes n}$, and a dyad can be viewed as a linear transformation (and hence a matrix) via the isomorphism $$\mathbb{R}^3 \otimes \mathbb{R}^3 \cong \mathbb{R}^3 \otimes (\mathbb{R}^3)^\ast \cong \operatorname{End}(\mathbb{R}^3).$$ The historical context you should keep in mind is that historically, determinants actually came long before matrices, and it wasn't until well into the 20th century that linear algebra really took the form we know today, with the emphasis on linear transformations (viz, matrices) as the primary object of interest. Even as late as the 1920's, matrices and matrix algebra were still so recherché, even in theoretical physics (!), that Heisenberg managed to independently "rediscover" matrix multiplication in the course of his work on quantum mechanics without knowing it, until Born (I think) pointed it out.

In any event, from what I understand, Reddy's use of dyadics and polyadics is not at all unusual for continuum mechanics, which presumably still uses them as an entrenched relic of an earlier time, not all that long ago, when they were the sensible, mainstream mathematical tools to use, and not linear transformations and tensors as we know them.

As for how to put a dyad in "nonionic form," I can only presume that it is given, in terms of the isomorphism above, by $$ {\bf e}_i{\bf e}_j = e_i \otimes e_j \mapsto E_{ij}, $$ where $\{e_1,e_2,e_3\}$ is the standard ordered basis for $\mathbb{R}^3$, and $E_{ij}$ is the matrix with entries $(E_{ij})_{kl} = \delta_{ij}\delta_{kl}$. However, do take a look at the relevant Wikipedia page.

Answer 2

First let's look at how to think of matrix transformations on ${\mathbf R}^3$. For any 3 x 3 matrix $A$, we can write the function $L({\mathbf x}) = A{\mathbf x}$ as a sum of 3 separate functions built out of the rows of $A$, using dot products.

Example. Suppose $$ A = \left( \begin{array}{ccc} 1 & 2 & 3\\ 4 & 7 & 2 \\ 9 & 2 & 5 \end{array} \right) = \left( \begin{array}{c} r_1\\r_2\\r_3 \end{array} \right), $$ where $r_1, r_2$, and $r_3$ are the rows of $A$. Then $$ A{\mathbf x} = (r_1 \cdot {\mathbf x})e_1 + (r_2 \cdot {\mathbf x})e_2 + (r_3 \cdot {\mathbf x})e_3. $$

This expresses the matrix transformation ${\mathbf x} \mapsto A{\mathbf x}$ as a sum of 3 linear transformations ${\mathbf x} \mapsto (r_i \cdot {\mathbf x})e_i$. The matrix way of writing $L({\mathbf x})$ is "nonionic" form (apologies to the chemists, but it doesn't mean "not ionic", but rather "nine-ish"), while the other way, as a sum of three terms with dot products, is the dyadic form.

For any two vectors $v$ and $w$ in ${\mathbf R}^3$ we can write down a linear transformation ${\mathbf R}^3 \rightarrow {\mathbf R}^3$ by the rule $L_{v,w}({\mathbf x}) = (v \cdot {\mathbf x})w$. The vectors $v$ and $w$ are fixed, while ${\mathbf x}$ varies. Such linear functions are not the most general linear functions from ${\mathbf R}^3$ to ${\mathbf R}^3$, since the values of $L_{v,w}$ are all scalar multiples of $w$ and thus lie along a line (which is not one of the standard axes if $w$ is not lying along an axis).

An example is $L_{e_1,e_3}$: $L_{e_1,e_3}(a_1e_1+a_2e_2+a_3e_3) = a_1e_3$.

Do you see the relation of matrix transformations with these dot product linear transformations $L_{v,w}$? We saw above how any matrix transformation can be written as a sum of three $L_{v,w}$'s, where the $w$'s are taken to be the standard basis. But we don't have to use the standard basis for the $w$'s. For example, we can just start with an $L_{v,w}$ where $w$ is not an $e_i$ and then write that linear transformation as a sum of such special functions with $w$ being an $e_i$.

For example, if $v = (2,1,0)$ and $w = (1,2,3)$ then for ${\mathbf x} = (a,b,c)$ we have $$ L_{v,w}({\mathbf x}) = (v \cdot {\mathbf x})w = (2a+b)w = \left( \begin{array}{c} 2a+b\\4a+2b\\6a+3b \end{array} \right) = \left( \begin{array}{ccc} 2&1&0\\4&2&0\\6&3&0 \end{array} \right) \left( \begin{array}{c} a\\b\\c \end{array} \right). $$ This last formula expresses $L_{v,w}$ as a matrix transformation, so by the same ideas as in the first example we have $$ L_{v,w}({\mathbf x}) = (r_1 \cdot {\mathbf x})e_1 + (r_2 \cdot {\mathbf x})e_2 + (r_3 \cdot {\mathbf x})e_3 $$ where the $r_i$'s are the rows: $r_1 = (2,1,0)$, $r_2 = (4,2,0)$, and $r_3 = (6,3,0)$. Thus $$ L_{v,w} = L_{r_1,e_1} + L_{r_2,e_2} + L_{r_3,e_3}. $$ Notice in particular that a single $L_{v,w}$ can be a sum of other $L_{v,w}$'s.

So far I haven't used any funky words like "dyad". I've shown by examples how any matrix transformation can be written as a sum of $L_{v,w}$'s.

Definition: A dyadic is just an $L_{v,w}$. A dyad is any sum of dyadics.

In concrete terms, a dyad is just a general linear transformation from ${\mathbf R}^3$ to itself, while a dyadic is a linear transformation whose image is one-dimensional (one of the $L_{v,w}$'s).

If you know what tensor products of vector spaces are, then a dyadic is the same thing as an elementary tensor in $({\mathbf R}^3)^* \otimes_{\mathbf R} {\mathbf R}^3$, where $({\mathbf R}^3)^*$ is the dual space of ${\mathbf R}^3$ (can be identified with ${\mathbf R}^3$ using the dot product). A dyad is a general tensor in $({\mathbf R}^3)^* \otimes_{\mathbf R} {\mathbf R}^3$. This tensor product can be interpreted as the collection of linear maps ${\mathbf R}^3 \rightarrow {\mathbf R}^3$, which is just the 3 x 3 matrices.

A polyad is a member of a tensor product of multiple copies of a vector space and its dual space. A polyadic is an elementary tensors in such a tensor product space. If you want to read a story about this terminology, see the last two paragraphs of https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf.