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Question involving Levi-Civita symbol

  1. Mar 28, 2013 #1
    Can someone please explain to me why

    [itex]\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial A_k}{\partial x_j} = 0[/itex]

    where A is a constant vector field.
     
  2. jcsd
  3. Mar 28, 2013 #2
    The double differentiation w.r.t the coordinates is symmetric: $$\frac{\partial}{\partial x_j}\frac{\partial}{\partial x_i}=\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}$$

    On the other hand, the Levi-Civita tensor is anti-symmetric on i,j. Interchange these indices and you therefore get: $$\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}=-\epsilon_{jik}\frac{\partial}{\partial x_j}\frac{\partial}{\partial x_i}=-\epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}$$
    where in the second equality we have renamed the dummy indices (they are summed over) i to j and j to i.
    So we get that something is equal to minus itself, and thus is zero.
     
  4. Mar 28, 2013 #3
    Makes perfect sense, thank you!
     
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