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Re: Representing a vector field with two scalar fields

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  • sxsw@rocketmail.com
    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:
      >
      > \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!
      >
    • Bedros Afeyan
      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 2 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:
        > 
        > \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!
        >


      • sxsw@rocketmail.com
        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 3 of 6 , Apr 20, 2009
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          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!
          > > >
          > >
          > >
          > >
          >
        • sxsw@rocketmail.com
          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 4 of 6 , Apr 21, 2009
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            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:
            > > > >
            > > > > \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!
            > > > >
            > > >
            > > >
            > > >
            > >
            >
          • Bedros Afeyan
            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 5 of 6 , Apr 28, 2009
            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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