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My background in physics is not that solid, but I am following a lecture where the Electromagnetic field tensor $F^{\mu \nu}$ was introduced as $F^{\mu \nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}$, where $A^{\mu}$ is the vector potential.

What I do not see is that why is in the absence of an external field the derivative given by $$\partial_{\mu} F^{\mu \nu} = \partial^2 A^{\nu} - \partial^{\nu}(\partial_{\mu} A^{\mu})?$$

Did they use the equation of motion? Because my calculation using the product/chain rule leads to $$\partial_{\mu} F^{\mu \nu} = \partial_{\mu}\partial^{\mu}A^{\nu} + \partial^{\mu}(\partial_{\mu}A^{\nu}) - \partial_{\mu}\partial^{\nu}A^{\mu} - \partial^{\nu}(\partial_{\mu}A^{\nu})$$

and I do not know how two of those four terms should exactly cancel in order to get the equation above. I would appreciate any help. Thank you

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You shouldn't use the product rule, because $\partial$ is not a function; you don't differentiate it. Simply put a $\partial^\mu$ everywhere, remembering that derivatives commute among themselves.

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