A Understanding the Colon Notation:

[Integrals]

What does the colon notation, :, mean?

 

[fresh_42]

What does the colon notation, :, mean when it is inside an integral for weak solutions?

Can you be a bit more precise and descriptive?

 

[Integrals]

Can you be a bit more precise and descriptive?

Apologies for the confusion, is it correct to say that $$\mathbf{x}\otimes\mathbf{x}:y$$ is equivalent notation for $$\langle\mathbf{x}\otimes\mathbf{x}, y\rangle$$ where the latter denotes the inner product. Perhaps, this question was more elementary than I had meant...

 

[fresh_42]

I haven't seen the first one, and the second one calls for the question, what are you multiplying there?

 

[Integrals]

I haven't seen the first one, and the second one calls for the question, what are you multiplying there?

Locally integrable functions - where $$\mathbf{x}=(x_1({\cdot}), x_2({\cdot}), x_3(\cdot))$$

 

[WWGD]

How is this related to distributions?

 

[fresh_42]

Locally integrable functions - where $$\mathbf{x}=(x_1({\cdot}), x_2({\cdot}), x_3(\cdot))$$

If I build the $\mathbf{x} \otimes \mathbf{x}$ from this, I get a rank one $3\times 3$ matrix. How can this be multiplied by an $y$ via an inner product, wherever this is from?

 

[Haborix]

The colon can signify a double dot product (see the wikipedia page on Dyadics). It is still ambiguous what is going on here. What is $y$?