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Dyadic form

  1. Jan 8, 2008 #1
    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:
    [tex]
    \int dx f(x)
    [/tex]
    which I find confusing. Has it got something to do with the dyadic form?
     
  2. jcsd
  3. Jan 8, 2008 #2
  4. Jan 8, 2008 #3

    Mute

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    Homework Helper

    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 [itex]\mathbf{uv}[/itex] 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 [itex]\nabla \nabla[/itex] in the green's function [itex]\mathbf{G}(\mathbf{r},\mathbf{r}')[/itex].

    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 [itex]\nabla \nabla[/itex] is as well.

    If we write the [itex]ij^{\mbox{th}}[/itex] component of G as [itex]G_{ij}[/itex], then

    [tex]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})[/tex]

    where i,j = 1, 2, 3, corresponding to directions r_1 = x, r_2 = y, r_3 = z, and [itex]\delta_{ij}[/itex] 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.
     
    Last edited: Jan 8, 2008
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