Proving Q = gradP + Sr for Splitting Vector Field B

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In summary, the conversation discusses the decomposition of a divergenceless vector field B into toroidal and poloidal parts, and the existence of a scalar potential P. The steps for proving P's existence are outlined, and it is mentioned that the author of a paper has stated that Q does not necessarily have to be equal to gradP, but can have a more general form. The conversation concludes with the suggestion to learn about de Rham cohomology for a deeper understanding of this concept.
  • #1
Cunicultor
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The question is about the decomposition of any divergenceless vector field, B, in Poloidal and Toroidal parts. It says in one paper that I've been reading
"Since B is solenoidal, it can be split into toroidal and poloidal parts, BT and Bp: B=curl(Tr)+curlcurl(Pr)"

I cannot find the way of proving that the scalar potencial P must really exist, I mean, I cannot prove that if we have a vector field divergenceless like B we have to have a decomposition like that.


This were my steps
.div (B)=0 therefore B=curl(A)

.A is the vector potencial and can be decomposed in 2 parts, one parallel to r and other perpendicular to r,i.e., A=Tr+Qxr

.Then B comes like B=curl(Tr)+curl(Qxr)
.Now, if Q were irrotacional then Q=grad P and the thing was done(B=curl(Tr)+curlcurl(Pr))

But how can I prove Q is really Q=gradP?


I've sent an email to the author and she said that Q is not required to be Q=gradP but isntead it should have the more general form
Q=gradP +Sr
Ok, it works fine and I get the final result as I want.
But again, can I really write Q like Q=gradP +Sr?? It doesn't seem obvious for me...

If you could give me a hand on this I would apreciate a lot.
 
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  • #2
Do you want to know why this is true? I mean the serious reason behind the why? Learn about de Rham cohomology then, the de Rham complex is exact, and that tells you when things lie in the image of grad, say. You might want to explain what all the symbols mean as well: what is Sr, for instance?
 
  • #3
Sorry not making it clear since the beginning;

T,S,P are scalar functions

B,A,Q are vector functions

r stands for radial vector




Rham cohomology ? I never heard about it... I don't think it would required such a complex thing, anyway...
 
  • #4
de Rham cohomology explains these phenomena, there may be an elementary reason, but remember what you're doing requires you to be in R^3.

Perhaps recall the case of integration - it is only defined up to addition of a constant - that is something that differentiation kills off.
 
  • #5
But so, what do you think about it?

What's the reason to have that kind ov vector decomposition?!
 
  • #6
What is de Rham cohomology?..where we can get the information regarding the same.please let me know
 
  • #7
Bott and Tu's book, the name escapes me, is supposed to be good.

It is related to differential forms and there might be something in the differential geometry forum.
 

1. What is the significance of proving Q = gradP + Sr for Splitting Vector Field B?

The equation Q = gradP + Sr is an important result in vector calculus, as it helps to understand the behavior of vector fields. It shows that a vector field can be split into two components - a gradient component and a solenoidal component. This is useful in many applications, such as fluid dynamics and electromagnetism.

2. How is the equation Q = gradP + Sr derived?

To derive the equation Q = gradP + Sr, we use the fundamental theorem of calculus and the Helmholtz decomposition theorem. The fundamental theorem of calculus states that a vector field can be represented as the gradient of a scalar field. The Helmholtz decomposition theorem states that any vector field can be decomposed into a gradient component and a solenoidal component. By combining these two theorems, we can derive the equation Q = gradP + Sr.

3. Can you provide an example of a splitting vector field B?

One example of a splitting vector field B is the electric field generated by a point charge. The electric field can be decomposed into a gradient component, which represents the potential energy of the charge, and a solenoidal component, which represents the electric field lines. This splitting of the vector field helps us to understand the behavior of the electric field around the point charge.

4. How is the equation Q = gradP + Sr used in practical applications?

The equation Q = gradP + Sr is used in many practical applications, such as in fluid dynamics, electromagnetism, and image processing. In fluid dynamics, it helps to understand the flow of a fluid around an object. In electromagnetism, it helps to analyze the behavior of electric and magnetic fields. In image processing, it can be used to separate an image into its gradient and solenoidal components, which can be useful for noise reduction.

5. Is the equation Q = gradP + Sr applicable to all vector fields?

Yes, the equation Q = gradP + Sr is applicable to all vector fields. This is because the Helmholtz decomposition theorem states that any vector field can be decomposed into a gradient component and a solenoidal component. Therefore, the equation is not limited to any specific type of vector field and can be applied to all cases.

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