Hi Dr. Afeyan,

Yes, I am well aware of that. I totally agree that the field perpendicular to
$\hat{b}$ can be decomposed into
$\nabla\Phi + \nabla\times\vec{A}$ with one constraint
$(\nabla\Phi + \nabla\times\vec{A}) \cdot \hat{b} = 0$.

The question is whether we can represent it with two scalar fields,
in the form of $\hat{b}\times\nabla\Phi$ and/or $\hat{b} \times (\hat{b} \times
\nabla \Psi)$.

Thank you.

--- In harmonicanalysis@yahoogroups.com, Bedros Afeyan <bedros@...> wrote:
>
> Dear SXSW,
>
> Are you aware of the Helmholtz theorem on general decompositions of
> vector fields into potentials that are curl free and divergence free
> (sometimes called the fundamental theorem of vector calculus)? Here is
> a way in:
>
> http://farside.ph.utexas.edu/teaching/em/lectures/node37.html
> or
> http://en.wikipedia.org/wiki/Helmholtz_decomposition
>
> Dr. Bedros Afeyan Bonde Court Office
> (925) 417-0609
> Polymath Research Inc. Regus Office
> (925) 399-6161
> 827 Bonde Court Fax
> (925) 417-0684
> Pleasanton, CA 94566
> cell (925) 209-5539
>
>
> On Apr 20, 2009, at 10:55 AM, sxsw@... wrote:
>
> >
> >
> > Dear All,
> >
> > Let me ask a related question here:
> >
> > What are the characteristics/properties of a vector field that can
> > be expressed
> > as $\hat{b}\times\nabla\Phi$ and/or $\hat{b} \times (\hat{b} \times
> > \nabla \Psi)$ ?
> >
> > For example, any vector field that can be expressed as $\nabla\Phi$
> > has the property $\nabla\times\nabla\Phi=0$, so we can check the
> > validity of the expression by taking curl with that vector field.
> >
> > Are there such properties we can check for $\hat{b}\times\nabla\Phi$
> > and/or $\hat{b} \times (\hat{b} \times \nabla \Psi)$ ?
> >
> > Thanks!
> >
> > --- In harmonicanalysis@yahoogroups.com, "sxsw@" <sxsw@> wrote:
> > >
> > > Hi all,
> > >
> > > Suppose I have a known 3-D vector field $\hat{b}$, is it always
> > possible to express another vector field(Let's call it A) which is
> > perpendicular to this vector field in the following form:
> > >
> > > \begin{equation}
> > > \vec{A}=\hat{b} \times \nabla \Phi + \hat{b} \times (\hat{b}
> > \times \nabla \Psi)
> > > \end{equation}
> > >
> > > We can see the above representation certainly guarantees that $
> > \vec{A}$ is perpendicular to $\hat{b}$.
> > >
> > > If the answer is yes, how should one represent $\Phi$ or $\Psi$ in
> > terms of $\vec{A}$?
> > >
> > > If the answer is no, what is the criteria for such representation
> > to be appropriate?
> > >
> > > Thanks!
> > >
> >
> >
> >
>