Proving Q = gradP + Sr for Splitting Vector Field B

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Discussion Overview

The discussion revolves around the decomposition of a divergenceless vector field, B, into poloidal and toroidal components, specifically addressing the existence and form of a scalar potential P in the context of vector fields. Participants explore theoretical aspects, mathematical reasoning, and implications of the decomposition.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to prove that a divergenceless vector field B can be decomposed into a form involving a scalar potential P, specifically whether Q can be expressed as Q=gradP + Sr.
  • Another participant suggests that understanding de Rham cohomology could provide insight into the conditions under which such decompositions hold.
  • Clarifications are made regarding the definitions of the symbols used, including the nature of the scalar functions T, S, P, and the vector functions B, A, Q.
  • There is a mention of the need for the discussion to remain within the context of R^3 for the decomposition to be valid.
  • Participants express uncertainty about the necessity of complex mathematical frameworks like de Rham cohomology for understanding the decomposition.
  • One participant refers to a book by Bott and Tu as a potential resource for further understanding the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity or form of the scalar potential P in the decomposition of the vector field. Multiple viewpoints are presented regarding the mathematical underpinnings and the complexity of the concepts involved.

Contextual Notes

There are unresolved questions regarding the assumptions needed for the decomposition, the definitions of the involved functions, and the implications of using advanced mathematical concepts like de Rham cohomology.

Who May Find This Useful

This discussion may be of interest to those studying vector calculus, differential geometry, and mathematical physics, particularly in the context of fluid dynamics or electromagnetic theory.

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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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?
 
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...
 
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.
 
But so, what do you think about it?

What's the reason to have that kind ov vector decomposition?!
 
What is de Rham cohomology?..where we can get the information regarding the same.please let me know
 
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.
 

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