I made inquiries about Helmholtz Decomposition to some Doctor. Here is

He is kind to answer my question as shown below,

I feel very weird in his sentence “This is referring to an abstract mathematical orthogonality of the curl and divergence operators, and not to a literal geometric orthogonality of the vectors.” .

These idea may be popular among hydrokinetic scientists, or among meteorologists.

But, I think these idea is infantile mistakes which is out of the basic mathematical vector concepts.

It’s Ok to think about an abstract mathematical orthogonality of the curl and divergence. But you have to check the orthogonality of them goes clear the orthogonality in the real world.

I think the components of vector is mistakenly thought among the hydrokinetics. “The components” are given at a point. Forexample, please see next page.

But, “the components in Helmholtz Decomposition” should be given in small space, not at a point. They should be determined in small space, or with neighbor vectors.

The basic mistake among hydrokinetics is that they think the non-divergence components and irrotational components are given at a point. They think that they can get these components at appoint by using delta functions and Fourier transformations. But, the “components” in Helmholtz Decomposition are not innate concepts such as “the vertical components and the horizontal component”. So, they should be given after calculation of “the divergence field and the curl field” in very small erea, but not at a point.

To determine the differential value at a point is the same mistake as Zeno paradox. One of Zeno paradox says that the flying arrow is stopping. You should think the arrow is flying or not, with velocity. But if you infinitely shorten the time of the arrow flying, and you can think the arrow is stopping at each second.

To be stopping or not should be considered by velocity, or “the length”/”the time”.

To shorten the time infinitely is Ok as long as time interval is not 0, for thinking the arrow is flying or not.

I don’t agree with using the ward “components” in Helmholtz Decomposition, but in this article, use it according to commonly used.

The non-divergence components and curl components are given as bellow

The divergence field and the curl field can be given by calculating from any vector field without Helmholtz Decomposition. But, these two field are not guaranteed to separately exist. If you just had mistook these field separately exist, you can get a scalar potential and a vector potential.

But if you want to prove Helmholtz Decomposition, you should prove that there is no component that contribute to both the divergence and curl composition.

If any vector has a component which contribute to both 0f the divergence component and curl component,

But, if any vector has not the component which contribute to both components,