High School Proof of Specific Covariant Divergence

[gerald V]

If the comma means ordinary derivative, then $(A_\mu A_\nu^{,\nu} - A_\mu^{,\nu} A_\nu)^\mu = A_\mu^{,\mu}A_\nu^{,\nu} + A_\mu A_\nu^{,\nu,\mu} - A_\mu^{,\nu,\mu} A_\nu - A_\mu^{,\nu}A_\nu^{,\mu} = A_\mu^{,\mu}A_\nu^{,\nu} - A_\mu^{,\nu}A_\nu^{,\mu}$, where $A$ is some vector field. Does the same hold if the comma means covariant derivative (then usually denoted as semicolon)?
Please forgive me for this simple question. Due to my considerations the answer is yes, but I don't trust in this completely.

 

[vanhees71]

Yes, because the product rule holds, and you've nowhere interchanged the order of differentiation, which holds for partial but not for covariant derivatives (if the space is not flat).

 

[gerald V]

Thank you very much. But I did change the order of differentiation in the third term to make it cancel against the second. First, I just renamed the indices $A_\mu^{,\nu,\mu}A_\nu \equiv A_\nu^{,\mu,\nu}A_\mu$, but then I replaced $A_\nu^{,\mu,\nu} \rightarrow A_\nu^{,\nu,\mu}$. So I now assume the relation does not hold if the derivatives are covariant.