[fruitkiwi]

Dear all,

I recently exchange idea with a research scholar on helicity, I write my helicity in the following
Helicity= (V dot ω)/(|V||ω|)

where V represents the velocity field, and ω is the vorticity determined through ∇ × V

he commended that my expression of helicity is confusing, he said "I strongly recommend using different symbols for the multiplication between real values and vectors. A scalar product between two vectors, i.e. a and b, can be also denoted as a^T b."

Can anyone here help me out on how should I rewrite my equation to answer his comment?

[LuckyTran]

What makes the dot product notation appropriate in this specific case, is you only have vectors and no tensors are involved. However, it becomes contextually ambiguous very quickly if, let's say, you have another tensor equation in the very next line.

This equation is fine, but all your other equations might need to change.

Also transpose of a vector means nothing unless you are invoking matrix operations and there is nothing here to suggest there is or is not any matrix being used to represent any vector/tensor. Again, more context is needed, preferably the full body of text that you are actually arguing over.

[fruitkiwi]

Originally Posted by LuckyTran

What makes the dot product notation appropriate in this specific case, is you only have vectors and no tensors are involved. However, it becomes contextually ambiguous very quickly if, let's say, you have another tensor equation in the very next line.


This equation is fine, but all your other equations might need to change.

Thannks Lucky for the checking, I noted it.

[hunt_mat]

Originally Posted by fruitkiwi

Dear all,

I recently exchange idea with a research scholar on helicity, I write my helicity in the following
Helicity= (V dot ω)/(|V||ω|)

where V represents the velocity field, and ω is the vorticity determined through ∇ × V

he commended that my expression of helicity is confusing, he said "I strongly recommend using different symbols for the multiplication between real values and vectors. A scalar product between two vectors, i.e. a and b, can be also denoted as a^T b."

Can anyone here help me out on how should I rewrite my equation to answer his comment?

I've not worked on helicity a great deal but the main point is that it's a topological invariant, for that you need to integrate over the region of interest, which is normally R^3.

Secondly, you've normalised it for some reason, can you tell me why you've done that? What you have is cosine of the angle between the two vectors.

[FMDenaro]

A correct matrix notation would imply the transpose notation, for example in the
Momentum equation the rigorous notation would be with the transpose me nabla operator for the divergence of the fluxes that, being tensors, need a matrix notation for the scala product.

But such boring notation is not used in practice