What is a Dyadic Form and What Does it Mean?

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In summary, the dyadic form is used to write integrals involving two vectors, and it has something to do with the Kronecker delta.
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daudaudaudau
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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?
 
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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.
 
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1. What is a Dyadic Form?

A dyadic form is a type of structure or pattern commonly used in scientific research, particularly in the fields of psychology and sociology. It involves the analysis of relationships between two individuals, known as dyads, and is used to study various aspects of human behavior and social interactions.

2. How is Dyadic Form different from other research methods?

Dyadic form differs from other research methods, such as surveys or experiments, in that it focuses specifically on the relationship between two individuals. It allows for a more in-depth understanding of the dynamics and interactions between individuals, rather than just collecting data on individual behaviors or opinions.

3. What are some common applications of Dyadic Form?

Dyadic form can be used in a variety of research settings, including studying romantic relationships, family dynamics, friendships, and even workplace interactions. It can also be applied in clinical settings to understand therapist-client relationships or doctor-patient interactions.

4. How is data collected in Dyadic Form research?

Data in dyadic form research is typically collected through various methods, such as observations, interviews, and self-report measures. Researchers may also use technology, such as video recordings or social network analysis, to gather data on the interactions between dyads.

5. What are the benefits of using Dyadic Form in research?

Dyadic form allows for a more detailed and nuanced understanding of the dynamics between two individuals, which can provide valuable insights into various aspects of human behavior and relationships. It also allows for the examination of complex social processes that may not be easily captured by other research methods.

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