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Newbie-changing the dot product to simple multiplication

  1. Oct 24, 2013 #1
    How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following:
    eb.[itex]\partial[/itex]cea=-[itex]\Gamma[/itex]a bc

    How do I get just an expression for [itex]\partial[/itex]cea?
     
  2. jcsd
  3. Oct 24, 2013 #2
    Here [itex]\Gamma[/itex]a bc = ea.∂ceb
     
  4. Oct 24, 2013 #3

    mathman

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    I have no knowledge of the particular symbols. However if you have the dot product of two vectors equal to a scalar, you cannot get one of the vectors from the scalar without further information. It is not enough just to know the other vector.
     
  5. Oct 24, 2013 #4
    The partial derivative of the coordinate basis vector eb with respect to the spatial coordinate xc is a vector, which can be expressed at a given point as a linear combination of the coordinate basis vectors:

    [tex]\frac{\partial e_b}{\partial x^c}=\Gamma^j_{bc}e_j[/tex]

    The [itex]\Gamma 's[/itex] are the components of the vector. If we dot this equation with the duel basis vector ea, we get:
    [tex]e^a\centerdot\frac{\partial e_b}{\partial x^c}=\Gamma^a_{bc}[/tex]

    The trick is to figure out how to represent the [itex]\Gamma 's[/itex] in terms of the partial spatial derivatives of the components of the metric tensor.
     
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