Problem
In Apostal's calculus volume 2 , there is an example which shows that a solenoidal vector field that is not a curl. Example states that proof is difficult at this stage . Can anyone please me some understanding why this can happen. That is on what kind of open sets a solenoidal vector field is always a curl of some other vector field in that set?
NB-Currently a sophomore .
I suppose that a solenoidal field is defined as a field whose divergence is null.
The Poincaré Lemma says that a divergence-free field is the curl of some vector field only if it is defined on a contractible set. ( You can see : What does it mean if divergence of a vector field is zero? )
A classical example is the field:
$$
\bf {V} = \left(\frac{x}{\left(\sqrt{x^2+y^2+z^2}\right)^3},\frac{y}{\left(\sqrt{x^2+y^2+z^2}\right)^3},\frac{z}{\left(\sqrt{x^2+y^2+z^2}\right)^3} \right)
$$
that is divergence-free but not a curl of any vector field.
