What is a Dyadic Form and What Does it Mean?

[daudaudaudau]

Hi.

Can anyone explain to me what a dyadic form is? It is used here for example: http://en.wikipedia.org/wiki/Electric_field_integral_equation.

I also have an older book where the author sometimes writes integrals like this:
$$\int dx f(x)$$
which I find confusing. Has it got something to do with the dyadic form?

 

[daudaudaudau]

Regarding my second question, I found this: http://mathforum.org/library/drmath/view/63721.html.

So now I just need to know what a dyadic form is :)

 

[Mute]

I very vaguely remember a chapter about dyadic forms in my vector calc textbook, but we didn't cover it in class and I didn't really read it closely, but it had to do with giving meaning to expressions like $\mathbf{uv}$ where u and v are both vectors and they are not in a dot product or cross product or anything - the result is actually a matrix, or some sort of rank 2 tensor.

On that page the "dyadicness" is due to the term $\nabla \nabla$ in the green's function $\mathbf{G}(\mathbf{r},\mathbf{r}')$.

There is a wikipedia page on the dyadic tensor: http://en.wikipedia.org/wiki/Dyadic_tensor

I don't know if that's very enlightening. It's not entirely clear to me - I have a better understanding of what a dyadic product is, the article of which is linked to at the bottom of that page, but I haven't looked closely enough to discern if they are essentially the same thing. I suspect they are, or are at least closely related, as the G above is a matrix, so $\nabla \nabla$ is as well.

If we write the $ij^{\mbox{th}}$ component of G as $G_{ij}$, then

$$G_{ij}(\mathbf{r}, \mathbf{r}^{\prime}) = \frac{1}{4 \pi} \left[ \delta_{ij} + \frac{\partial_i \partial_j}{k^2} \right] G(\textbf{r}, \textbf{r}^{\prime})$$

where i,j = 1, 2, 3, corresponding to directions r_1 = x, r_2 = y, r_3 = z, and $\delta_{ij}$ is the Kronecker delta, which is equal to 1 if the indices are equal and zero otherwise. The i here refers to the row of the matrix, and the j refers to the column (I think I have that in the right order).

Note that here I paid no attention to the difference between covariant and contravariant vectors, as if I did technically the above thing wouldn't be a matrix but some other sort of rank two tensor, so one index should probably be a superscript, but that's probably getting beyond what you know or care about at the moment.