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The right way to prove Helmholtz Decomposition

May 20, 2017

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

From → Meteorology

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