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About the components of vectors in Helmholtz Decomposition

If you deal with a vector field, the components of vector is given at a point. Generally, the components of vector are axial components in the coordinate system you’re dealing with

 But, the components in Helmholtz Decomposition theorem mean how does nabladotFthe vector contribute to divergence and curl in the small neighborhood, not at a point. That is not given by only one vector at the point.

 Although, many scientists say that divergence and curl can be given at a point, they are never given at a point. Their mistake is the same one with “Zeno paradox”. They say that “The flying arrow is rest”.

ZenoParadox

Some scientist was kind to teach me that if we can get the normal component and tangential component on the boundary, we can get the scalar potential and vector potential. Why can we get them without getting the divergence and curl on that point.

If the contribution of the vector to “divergence ” and “curl ” is considered as “components”, you should consider other 2 kinds of components. They are “non-divergent and irrotational component” and “divergent and rotational component” outside of “divergent component” and “ rotational component”.

4 kinds of vector components

I can’t believe that scientists have been satisfied with just two components about this phenomena.

Taking 4 kinds of components, I will show you some example of vector fields.


(1) the uniform stream

You can make sure the existence of the component which does not contribute to divergence nor to curl to think about uniform stream.

For example, we can consider a uniform stream run eastward with 5 knots.

uniform_flow

This flow has a scalar potentialfai, because it is irrotatinal. We can considerfai=5x+C. The flow vector F=nablafai=5i. Then the divergence nabladotF=0.

And this flow has a vector potential A , because it is non-divergent. We can consider A =5yk(upward unit vector). Then, F=A_curl=5i. Then the curl F_curl=0.

component of uniform stream

According to Wikipedia, Helmholtz Decomposition says like below,

Helmholtz_equation_wiki.png

But, nablafai=0 andA_curl=0 for the uniform flow.

Although, Helmhpltz Decomposition says that any vector F can be decomposed into divergence component and curl component, but if the vector field is uniform flow, it is not true.

 

(2) electric field

There is a electric field around a charged particle. If other unit charge is put in that field, it is forced outward or inward to the charged particle. At any point any charg is forced, and they make a force vector field.

This force vector field is composed of 100% divergence component as shown below.

electric field2.png

 

(3) magnetic field

Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force Law.

 magnetic force

According to Maxwell’s equations, the corresponding formula for magnetic fields: So, the divergence of B is always 0.

This magnetic force vector field is composed of 100% curl component as shown below.

component of rotatinal field


(4) generally, liquid or gas flow has all components of 4 kinds

The flow of liquid or gas has generally all 4 kinds of components.

Because, divergence is given by nabladotF、and curl is given by F_curl.

Or, in spherical coordinats, divergence and curl are given as below,

divergence and curl with polar coordinate_t

From these equations, I think the vector F basically contributes to both of divergence and curl components . Because, divergence and curl are given by the same vector F.

In the early time of usage of the differential operator nabura, Maxwell, Tait and Helmholtz and others think about electromagnetic phenomena, I think.

There is big difference between the vector fields of electromagnetic wave and the flow vector field of liquid. There certainly exist the scalar potential for electro field, and the vector field for magnetic field. But scalar potentials and vector potentials of the flow vector of liquid or gas do not exist in the tangible form.

If there is a vector field F, you can calculate divergence field nabladotF and curlfield F_curl. But they might be dependent on each other. If they have some intersectional parts like shown below

the components of general fluid atream

If F has the component that contribute to both divergence and curl, F can’t be decomposed into divergent components and curl components.

If you cling onto Helmholtz Decomposition, you should prove that F has not the component which contribute to both divergence and curl.

 Or right from the start, is there not the component which contribute to both divergence and curl in the flow vector field, not in the electromagnetic wave phenomena.

 

 

 

Image

components and contributions

components and contributions

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

my quation to Dr Baird

He is kind to answer my question as shown below,

the responce from Dr. Baird

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.

01_a vector and their conponents

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

02_contributions

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,

Fig for F has both contribution

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

Fig for F has not both contribution

Any vector contribute to both the divergence role(component) and curl role(component)

 

Acording to Wikipedia, Helmholtz Decompose theorem states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field

I want to show you a simple example in two dimension cooerdinates system as below figure.Helmhltz_Decomposition_by_difference_approach

Fig 1 Calculus equation for the divergence and the curl

The divergence of vector F at a point o is calculated by using blue vectors, or V1, U2,V3 and U4 in the figure above , and the curl of vector F is calculated by using red vectors, or U1,V2,U3 and V4.

All of vectors which are decomposed from vector F for calculating the divergence and curl have been used.

So, you can say this is Helmholtz Decomposition.

 

Certainly, you can think about an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field with any vector field.

 

But if vector F has some component that contribute to both vector field, you can’t say that the vector field decomposed into an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field as showing bellow.

Fig for F has both contribution

Fig 2 If F(r) has some component that contribute to both

If vector F is smooth, U4 bears distinct relationship to U3 in Fig 1.

If_A_is_differentiable5

Fig 3 the conditions for smooth vector field

 

U4 that is constitutive part of divergence is given by a equation include U3 that is constitutive part of curl.

So, you can say that Any vector field has some component which contributes to both  roles(components) of divergence and curl.

Therefore, Helmholtz Decomposition is wrong.

The right way to prove Helmholtz Decomposition

Acording to Helmholtz Decomposition theorem, a vector can be decomposed into a curl-free component and a divergence-free component.

A vector and its components

What is a vector?  A vector is a quantity that has both magnitude and direction.

Vectors in multiple-dimension coordinate systems can be broken up(=decomposed) into their component vectors. In the three-dimensional case, this results in a x-component, a y-component and a z-component . The next picture is an example of a Force vector (F) broken into its components (Fx and Fy) in the two-dimensional case. When breaking a vector into its components, the vector is a sum of the components.

The components are broken up from a vector at just a point.

Fig of components of a vector

The components generally mean the elements which are broken up (=decomposed) from a vector in Cartisian coordinate system. Each component is independent to each other.

       The components in Helmholtz Decomposition

But the components used in Helmholtz Decomposition are not the same components of vector in Catisian coordinate system. They represent spatial aspect in a vector field.

I think that the “components” in Helmholtz Decomposition are rather the “roles” of a vector within the neighbors for divergence and curl. They should be decided within neighbor vectors.

Please, think about a vector field without Helmholtz Decomposition. Thare is just a vector field F at first.

So you can calculate and get the divergence field by nabra_dotF, and the curl field by nabra_productF. Helmholtz Decomposition is not needed to calculate these calculation.

The divergence field made bynabra_dotF has an irrotational vector by nabla(nabra_dotF), and the curl field made by ∇×F has a non-divergence vector by nabra_product(nabra_productF) .

An irrotational vector field has a scalar potential chi=∇・F, and  a non-divergence(solenoidal) vector field has a vector potential A×F

In Meteorology, chi is called velocity potentials, and A is called stream functions.

Here, you have gotten the divergence vector field nablachi which is curl-free, and the curl vector field nablacrossA which is non-divergence from a vector F.

 Many articles about Helmholtz Decomposition are similer to these expositions above.

But, you have never guaranteed that the sum of these two vectors make up the original vector, or the original vector F can be decomposed into these two of vectors. You have just gotten an irrotational vector and non-divergence vector from F, but the sum of two may not 100% of F. Because, these elements, or  a curl-free component and a divergence-free component are not the components in Cartisian coordinate system.

The original vector F may have the rest of them. Or the vector F might have contribute to both “components”( I want to say them “roles”).

If the vector included some components( or roles) which contributes to both of curl components and divergent components, the original vector could not perfectly decomposed into a curl-free component and a non-divergence component.

Fig for F has both contribution

 

      the right way to prove Helmholtz Decomposition 1

For all that, I can prove that Helmholtz Decomposition theorem is right.

I will show you it with diagram bellow.

Helmhltz_Decomposition_by_difference_approach

I want to get the divergence and curl at a point (r), at the center of figure above. Therefor, I need F2 and F4 to get differential value of F with x axis, and F1 and F3 to get the differential value of F with y axis. F1,F2,F3 and F4 act for 100% of F(r) to calculate the differential of F at this point

And each of F1,F2,F3 and F4 can be decomposed into x components and y components in the cartesian coordinate system. They are U1 and V1 from F1, U2,V2 from F2, U3 and V3 from F3, U4 and V4 from F4.

Therefor, U1,U2,U3,U4,V1,V2,V3 and V4 act for 100% of a vector F at this point for calculation of space derivative. And half of them that is V1,V3,U2 and U4 contribute  only to divergence, and another half of them that is V4,V2,U3 and U1 contribute only to curl.

So, we can say that we use all of differential element of F and use half of them to calculate the divergencenabra_dotF, and use another half of them to calculate the F_curl. So, The vector F can be decomposed intonablanabra_dotF) and nabra_productF_curl). This is the perfect proof for Helmholtz Decomposition.

the right way to prove Helmholtz Decomposition 2

I know that there is another way to prove Helmhltz Decomposition thorem. If you can find an identity that  shoes some vector function equal to the sum of non-divergence vector and curl-free vector, it is the another way that can prove Helmholtz Decomposition.

triple_bector_multiplication

The difference between differential operators and vectors

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction.

It can be expressed like

the general form of a vector

based on three perpendicular spatial axes generally designated x, y, and z. Here, i,j,k are the unit vectors in the coordinate system respectively.

While, the differential operator “nabra” is similarly expressed as well as a vector.

nabra

So, this can be thought as a kind of vector.

But, we need to know there is big differences between a general vector and this differential operator, “nabra”.

 

A general vector is defined at a point. But, differential operator “nabra” is defined in the small area, not at a point.

Many scientist neglect this differences. For example, there is an article about Helmhltz Decomposition in Wikipedia like bellow

Part_Longitudinal and Transverse fields

In the upper quoted article, the first(upper) group of equations is OK, because they are expressed at a point. But the second(under) group of equations is not allowed.

We can easily get a sample flow which has divergences by transvers components and rotations by longitudinal components. They are not 0.

For example, let us think a flow like following Figure.

a flow with both components

The sample shows that a flow field is squared off at 50m intervals. And we can get the velocities at each grid point. Their values are shown in the Table.

So, we can get two components at any grid. They are the longitudinal component and transverse component.

transvers and longitudinal components

And, we can calculate the divergence with transverse components, and rotation with longitudinal components.

A sample flow with curl and divergence

 

They both may not be 0.

The mistake in Wiki’s article is caused by the neglect of the differences between the general vector and the difference operator.

 

Someone might say that you could decompose a vector into two components which are the longitudinal component and transvers component at the center point by using Fourier transformation and Invers Fourier transformation.

But, I want to say that the differential is not given at a point even if using Fourier transformation and invers Fourier trans formation. It is given in the very small area, but not at a point.

“True” actual components of a flow

We used to use isobaric charts for the analyses of weather phenomena.

For example, 200hPa charts show the situations on the plane of about 12000m height.

We can consider the wind data on 200hPa as a flow vectors in two-dimensions.

JMA and NOAA are making “velocity potentials” and “stream functions” from these wind data sets by applying “Helmholtz Decomposition”.

But I think that Helmholtz Decomposition is wrong, so these velocity potentials and stream functions are made by their mistakes.

I want to show you that how these potentials are made.

1) On the Helmholtz Decomposition theorem

There is a theorem called Helmholtz Decomposition that says any flow can be separated into irrotational divergent flow and non-divergence rotational flow. And velocity potentials can be calculated from the irrotational flow and steam functions can be calculated from the non-divergence flow.

JMA and NOAA are publishing velocity potentials and stream functions on net.

Acording to Helmholtz Decomposition, two kinds of these potentials are independent from one another. So, if you wanted to analize the distributions of divergence in some layer, you could do it by just analyses of velocity potential map.

From Wikipedia, Helmholtz Decomposition is given as follows
Equation1 ・・・・・・・・・・・・・1)

Equation2・・・・・・・・2)

Equation3・・・・・・・3)

Here, Fl means irrotational divergent flow, and Ft means non-divergence rotational flow.

I show you Fig.1 to image the Helmholtz Decomposition..

Fig01
Fig.1 Illustration for Helmholtz Decomposition

2) The third component

When we think about composition of vector, that is generically considered as projections of a vector which is given at one point onto the reference axes.

But the components in the Helmholtz Decomposition are given as roles which play as a flow in the set of neighbor flows. So, you should think about the component which play both role of curl and divergence, over and above the irrotational divergence component and the non-divergent rotational component.

If there is the third component which play both roles of curl and divergence in any flow, the components of any flow should be shown as Fig.2.

Fig02 Fig.2 The components of any flow

Even if there is the third component, you can calculate nabra_dotF and ∇×nabra_productF distributions, and therefor you can get velocity potential and stream function. And furthermore, you can get Fl and Ft.

But as you can see inFig.2, the composition of these two components does not match the original flow.

So, I can say that Helmholtz Decomposition is wrong.

3) the components of a “true” actual flow

Actually, there is a fair percentage of non-divergence and irrotational component in actual flow.

So, when you divide a flow into some components, you should think about the fourth component which has neither divergence nor curl(rotation).

Fig03 Fig.3 the 4 components of a general flow

 

According to Equation 2), φ is calculated from the term of nabra_dotF. So, Fl should consist of the components 1+3.

 Fig.4 is calculated with the components of 1+3

 Fig04

Fig.4 Fl is calculated with the component of

And, vector potential A is calculated with the term of nabra_productF.

So Ft should consist of the components ②+③.
Fig05Fig.5 Ft is calculated with the components of ②+③

 

Therefore, Equation 1) is not correct. Therefore we should say that Helmholtz Decomposition is wrong.
Fig06
Fig.6 Helmholtz Decomposition is not correct

After publishing this article, I should edit or remove my latest blog “On the components in Helmholtz Decomposition Theorem”, but I daringly keep it on the Net.

4) Another Decomposition

There is another way to decompose any wind into two components. You can decompose any wind into geostrophic wind component and ageostrophic wind component.

Geostrophic winds are theoretically given from the contours(heights of an isobaric surface). Geostrophic winds blow along contours in inverse proportion to the gap of contour lines. So, geostrophic winds are perfectly non-divergent wind.

And, because the natural winds blow as quasi-geostrophic winds, they mostly consist of geostrophic winds.

Ageostrophic winds are given as the difference calculated by subtracting geostrophic wind from the original(analyzed) winds.

So, there is no doubt in this way to divide any flow into geostrophic wind and ageostrophic wind.

I show you a illustration to image the decomposition which make a flow divided into geostrophic wind and ageostrophic wind inFig.7.
Fig07
Fig.7 The Decomposition into Geostrophic wind and ageostrophic wind

And, Fig.8 is an example for analyzed(original) wind(black arrow), geostrophic wind(blue arrow) and ageostrophic wind(red arrow).
Fig08
Fig.8 an example for geostrophic wind(blue), ageostrophic wind(red)

and analyzed wind(black) on 20th Jun 2011

Fig.8 shows that the composition of geostrophic wind and ageostrophic wind is nearly equal to original analyzed wind. It might be expected.

We can’t say that Geostrophic wind (blue arrow) take out 100% of the non-divergent component from natural(analyzed) wind(black arrow). But it mostly consist of them.

Ageostrophic wind component(red arrow) is approximately compounded of divergence component which is shown (1+3) in Fig.3.

Therefore, ageostrophic wind is nearly divergent wind which would be given from velocity potential.

Here I want to show the divergent wind and curl wind on the same day. The divergent winds were calculated from velocity potentials, and the curl winds were calculated from stream functions in Fig.9.
Fig09
Fig.9 an example for curl wind(blue), divergent wind (red)

and analyzed wind(black) on 20th Jun 2011

After seeing Fig.9, I had been left speechless for a while. Because the composition of divergent wind and curl wind is nearly equal to the original(analyzed) wind.

Had I mistook in former article related Fig.6?

Please put it aside for a while, and confirm that the divergent winds are fairly equal to ageostrophic winds.

This is an example for that Fl in Fig.4 is nearly equal to the component of 2 in Fig.7. We can say that ageostrophic winds are nearly equal to divergent winds.

5) about stream function

Comparing Fig.9 to Fig.8, curl winds Ft calculated from stream functions nearly equal to geostrophic winds.

And we have confirmed that the composition of divergent wind Fl and curl wind Ft is nearly equal to the original(analyzed) wind. This can be a proof for that Helmholtz Decomposition is crrect.

Here, I doubt if these stream function was truly calculated by using equation 3).

Please look at Fig.5 again.

Stream function must be driven from a vector potential expressed as below.
Equation3
Therefore, the components of just 2 and 3 in Fig.5 is useful to calculate nabra_productF. Because, even if the component 1 and 4 were used, they came to 0 as a consequence. So, Ft definitely not be nearly equal to geostrophic wind. Ft should be fairly small than geostrophic wind.

According to equation 2) and 3), the composition of Fl and Ft should be smaller than original analyzed wind.

There is a way to make these stream functions published from JMA or NOAA.

If you priliminaly beleaved Helmholtz Decomposition is right(Fig.1), you could get stream function from the difference calculated by subtracting divergent wind from the original(analyzed) wind.

But, there are 4 kinds of components in any actual wind.
Fig10
Fig.10 Actual way to get “Stream function”

It is all right to get Fl from the equation 2). But Ft must be calculated as the differences calculated by subtracting Fl from original wind, for getting equation 1). In any another way, Fl+Ft would not be equal to original F.

To take this way is definitely distinct from Helmholtz Decomposition. This way is the same way to separate a wind into geostrophic wind and ageostrophic wind.

Here, I must confess that I don’t know exactly how to make stream function. Please ask some person who know how to calculate the stream function, if you know. And ask him to publish the way how to calculate the stream function. I think it have been top-secret among them.