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!
>