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(adsbygoogle = window.adsbygoogle || []).push({});

"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 doesnt seem obvious for me...

If you could give me a hand on this I would apreciate a lot.

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# Spliting a Vector

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