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Denote the value of the determinant

  1. Apr 9, 2012 #1
    In "A Short Course in General Relativity", I met a statement that says if we denote the value of the determinant [itex]|g_{ab}|[/itex] by ##g##, then the cofactor of ##g_{ab}## in this determinant is ##gg^{ab}## and following this we can deduce ∂##_cg=####(##∂##_cg_{ab})gg^{ab}##.

    First, I don't understand what the first argument is.
    Second, if the first argument is true, I don't know how to derive the second one.

    Any help would be appreciated!
     
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2
    Do you know what a cofactor of a matrix is?

    Do you know how to relate the inverse matrix to the cofactors?

    Given the Laplace expansion:
    [tex]
    g = \sum_{a, b}{g_{a b} \, \mathrm{Cof}_{a, b}}
    [/tex]
    what element(s) of this sum contain the term [itex]g_{i j}[/itex] for fixed indices i and j?

    Then, what would:
    [tex]
    \frac{\partial g}{\partial g_{i j}} = ?
    [/tex]
    be?

    Knowing these partial derivatives, w.r.t. every element of the metric tensor, can you write the total differential of the determinant g?

    This is a very important step in studying GR and you should remember it.
     
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