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• ## Re: Representing a vector field with two scalar fields

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• Dear All, Let me ask a related question here: What are the characteristics/properties of a vector field that can be expressed as $hat{b} times nabla Phi$
Message 1 of 6 , Apr 20, 2009
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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:
>
>
> \vec{A}=\hat{b} \times \nabla \Phi + \hat{b} \times (\hat{b} \times \nabla \Psi)
>
>
> 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!
>
• 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
Message 1 of 6 , Apr 20, 2009
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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:

or

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:
>
>
> \vec{A}=\hat{ b} \times \nabla \Phi + \hat{b} \times (\hat{b} \times \nabla \Psi)
>
>
> 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!
>

• Hi Dr. Afeyan, Yes, I am well aware of that. I totally agree that the field perpendicular to $hat{b}$ can be decomposed into $nabla Phi + Message 1 of 6 , Apr 20, 2009 View Source Hi Dr. Afeyan, Yes, I am well aware of that. I totally agree that the field perpendicular to$\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}$, is it always > > possible to express another vector field(Let's call it A) which is > > perpendicular to this vector field in the following form: > > > > > > > > > \vec{A}=\hat{b} \times \nabla \Phi + \hat{b} \times (\hat{b} > > \times \nabla \Psi) > > > > > > > > > We can see the above representation certainly guarantees that$
> > \vec{A}$is perpendicular to$\hat{b}$. > > > > > > If the answer is yes, how should one represent$\Phi$or$\Psi$in > > terms of$\vec{A}$? > > > > > > If the answer is no, what is the criteria for such representation > > to be appropriate? > > > > > > Thanks! > > > > > > > > > > • Dear All, I think I can put my original question in a cleaner, equivalent form: Given an arbitrary vector field in 3-D,$ vec{A}$, is it always possible to Message 1 of 6 , Apr 21, 2009 View Source Dear All, I think I can put my original question in a cleaner, equivalent form: Given an arbitrary vector field in 3-D,$\vec{A}$, is it always possible to decomposed it into the following form:$\vec{A}=\nabla\Phi + \hat{b}\times\nabla\Psi + \lambda\hat{b}$where$\Phi$,$\Psi$and$\lambda$are scalar functions in 3-D, and$\hat{b}$is some known vector field in prior. If one investigate the above vector equation component by component, it is really similar to asking whether a linear coupled PDE always have unique solution. I am very ignorant of the PDE theory and I would hope some experts in this area could give me some ideas or comments. I will be very grateful. --- In harmonicanalysis@yahoogroups.com, "sxsw@..." <sxsw@...> wrote: > > Hi Dr. Afeyan, > > Yes, I am well aware of that. I totally agree that the field perpendicular to$\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}$, is it always > > > possible to express another vector field(Let's call it A) which is > > > perpendicular to this vector field in the following form: > > > > > > > > > > > > \vec{A}=\hat{b} \times \nabla \Phi + \hat{b} \times (\hat{b} > > > \times \nabla \Psi) > > > > > > > > > > > > We can see the above representation certainly guarantees that$
> > > \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!
> > > >
> > >
> > >
> > >
> >
>
• Hello again, It may be illuminating to look at this 1957 paper by Chandrasekhar and Kendall on vector wave equation solutions and their relation to scalar wave
Message 1 of 6 , Apr 28, 2009
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Hello again,

It may be illuminating to look at this 1957 paper by Chandrasekhar and Kendall on vector wave equation solutions and their relation to scalar wave equation solutions (in the context of force free magnetic field evolution in fluids, thus involving no fluid motion) to further this discussion on scalars shedding curls, shedding divergences. I include the paper below. See in particular, eqns. 5 to 8.

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