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Decompostion of scalar field

  1. Apr 12, 2014 #1
    If a vector field can be decomposed how a sum of a conservative + solenoidal + harmonic field...
    imagem.png

    so, BTW, a scalar field can be decomposed in anothers scalar fields too???
     
  2. jcsd
  3. Apr 13, 2014 #2
    What types of scalar fields are there?
     
  4. Apr 14, 2014 #3
    How about
    $$\phi (q) = \frac{\phi(q) + \lvert \phi(q)\rvert}{2} +\frac{\phi(q) - \lvert \phi(q)\rvert}{2}$$
    for a point ##q \in Q## on a smooth manifold and a section ##\phi \in C^{\infty}(Q, Q \times \mathbb R)## of the trivial vector bundle?
    This is a decomposition into positive and negative parts.
    If the operation ##q \to - q## makes sense, then you can also take
    $$\phi (q) = \frac{\phi(q) + \phi(-q)}{2} +\frac{\phi(q) - \phi(-q)}{2}$$
    using the same trick. This is a decomposition into symmetric and anti-symmetric parts.
    EDIT: You might be interested in this: http://en.wikipedia.org/wiki/Hodge_decomposition
    Note that a "scalar field" is a ##0##-form since ##\bigwedge^0 T^*Q \simeq Q \times \mathbb R##.
     
    Last edited: Apr 14, 2014
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