Electromagnetic tensor and its components

Question

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.

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.