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Homework Help: Ecuations Resolution

  1. Jan 3, 2005 #1
    Dear Friends,

    Another questions about mathmatics.

    How can be resolved a ecuation like this?

    [tex]{ \frac{ \partial{(m \vec {v} - \Psi \vec {v})}}{ \partial {t} } = \nabla (\Psi \vec {v}^2 )[/tex]

    Asuming that v could be any vector, m a constant, and psi a wave function. It's not the similar that a wave ecuation.

    And more:

    Is this ok? [tex] \Psi (\vec {v} \nabla) \vec {v} = \Psi \vec {v} (\nabla \vec {v}) [/tex]
     
  2. jcsd
  3. Jan 3, 2005 #2

    dextercioby

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    Homework Helper

    So i should understand that the unknown from your equation would be [itex]\Psi [/itex] ??That is to say,all other quantites are known... :confused:
    So your equation should be looking like that
    [tex] -\vec{v}\frac{\partial\Psi}{\partial t}=\vec{v}^{2}\nabla\Psi+\Psi\nabla(\vec{v}^{2})-m\frac{\partial\vec{v}}{\partial t} [/tex]

    I'lm afraind your equation is not 'balanced'.It has only one unknown and three equations.It's actually a system of 3 differential eq.with partial derivatives,but still only one function.Now,if 'v' is an unknown vector function as well,then the eq.(the system of eq.is not 'balanced' again).This time it would 4 unknowns,but only three quations.

    My guess,it's something fishy with your eq.Double check it.


    Write it in tensor notation.Simplify through the (assumed nonzero) scalar function [itex] \Psi[/itex] and u'll get
    [tex] v_{i}\frac{\partial v_{j}}{\partial x_{i}} \vec{e}_{j} [/tex]
    is the LHS.
    [tex] v_{j}\frac{\partial v_{i}}{\partial x_{i}} \vec{e}_{j} [/tex]
    is the RHS.
    You can see they're different.The first is a contracted tensor product between the vector 'v' and its gradient (which is a second rank tensor),while the second is the product between the vector and its divergence (which is a scalar).Though 'balanced' wrt to tensor rank,the two sides are different.
    Therefore,u don't have an equality.

    Daniel.
     
    Last edited: Jan 3, 2005
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