Decomposition of a Divergenceless Vector Field

[Cunicultor]

Viva!


I usually come upon this statement:

" Since B is solenoidal, it can be split into Toroidal and Poloidal parts, i.e, B=Bt+Bp, where Bt=curl(Tr) and Bp=curlcurl(Pr)"


How can I prove this??

I think it is somehow related with the stokes theorem...


Looking forward for someone to explain me this, once for all.
Cheers!

 

[HallsofIvy]

We can't answer that without knowing what "Tr" and "Pr" are.

 

[tacman]

The decomposition of a divergenceless vector field is a fundamental concept in vector calculus and has many applications in physics and engineering. The statement you have mentioned is known as the Helmholtz decomposition theorem, which states that any vector field can be decomposed into two parts: a solenoidal (divergenceless) part and an irrotational (curl-free) part.

To prove this, we can use the Stokes' theorem, which relates the surface integral of a vector field to the line integral of its curl. Using this theorem, we can write the vector field B as:

∫∫S B·dS = ∫C (curlB)·dl

where S is a closed surface and C is the boundary curve of that surface.

Now, since B is divergenceless, we can apply the divergence theorem to the left side of the equation, giving us:

∫∫S B·dS = ∫∫∫V ∇·B dV

where V is the volume enclosed by the surface S.

Since B is divergenceless, we can rewrite the right side of the equation as:

∫∫∫V ∇·B dV = ∫∫∫V 0 dV = 0

Therefore, we have:

∫∫S B·dS = ∫C (curlB)·dl = 0

This implies that the line integral of the curl of B over any closed curve C is equal to zero. This can only be true if the curl of B is itself equal to zero, i.e. B is irrotational.

Now, using the Helmholtz decomposition theorem, we can write the vector field B as:

B = Bt + Bp

where Bt is the toroidal part, given by Bt = curl(Tr), and Bp is the poloidal part, given by Bp = curlcurl(Pr).

Since the curl of B is equal to zero, we have:

curlB = curl(Bt + Bp) = 0

This implies that both Bt and Bp are also curl-free, i.e. irrotational.

Therefore, we have shown that any divergenceless vector field can be decomposed into two parts: an irrotational part and a solenoidal part. This proves the statement you have mentioned and also shows the relation between the Hel