I'm dealing with the covariant formulation of electromagnetism and I've come across the Electromagnetic tensor after learning a bit about the covariant notation. In particular I've problems understanding this passage in the file I'm studying in which the identity of the electromagnetic tensor components and the electromagnetic fields is proven. I've come across the following:

\begin{equation} F^{ij} = \partial^{i} A^{j} - \partial^{j} A^{i} = (\delta^{i}_{m} \delta^{j}_{n} - \delta^{j}_{m} \delta^{i}_{n}) \partial^{m} A^{n} = \epsilon^{ijk} \epsilon_{mnk} \partial^{m} A^{n} = - \epsilon^{ijk} (\nabla \times \vec{A} )_k \end{equation}

Now, I understand all the passages up to the change from the Levi Civita symbol to the curl: I know the curl can be written that way but I don't understand where the minus sign come from, I've thought that it could come from the relation $ \partial^{i} = -\partial_{i}$ for the spatial derivatives but I mess up with the position of the indices. I think mine is a lack of understanding of the covariant formulation and in particular of the definition of the curl with the correct position of indices. Thanks to whoever might want to answer.


1 Answer 1


You are perfectly right. The gradient symbol $\nabla$ represents the derivative $\partial_i$ with the index down, and the vector $\vec{A}$ naturally has components $A^i$ with the index up. Therefore:

$$\epsilon_{mnk} \partial^m A^n = \epsilon_{kmn} \partial^m A^n = - \epsilon_{kmn} \partial_m A^n = - (\nabla \times \vec{A})_k$$

Of course, this is all assuming we're using the $(+\ -\ -\ -)$ convention for the metric. If we're using the opposite convention, there's no minus sign.

  • $\begingroup$ This was my first intuitive idea but what I don't understand is the position of the indices: ther's the contraction between the n in the $\epsilon_{kmn}$ and the one in $A^{n}$ but what about the m, I don't understand the indices contraction convention in this peculiar case $\endgroup$ Jul 25 at 18:05
  • 1
    $\begingroup$ We're working in 3D space with Cartesian coordinates (really, a 3D subspace of 4D spacetime). This means that, as long as we're careful about changing signs when we move indices up and down, we can extend the summation convention to all repeated indices, even if they are both upstairs or both downstairs. This is because we only need our formulas to be invariant under rotations, not under Lorentz or more general transformations; and rotational invariance is maintained even with a contraction like $\epsilon_{kmn} \partial_m$. $\endgroup$
    – Javier
    Jul 26 at 14:25

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