Are there differences in notation for the d'Alembert operator?

Question

On Wikipedia the d'Alembert operator is defined as
$$square = partial ^alpha partial_alpha = frac{1}{c^2} frac{partial^2}{partial t^2}-nabla^2 $$
However, my professor uses the notation:
$$ square = partial _alpha partial^alpha$$
with $partial_alpha = frac{partial}{partial x^alpha}$ and $partial^alpha = frac{partial}{partial x_alpha}$ 
Is there a difference or are both notations equivalent

Jakob · Accepted Answer

As it was pointed out in the comments, both expressions are the same: Note that
$$partial^alpha = eta^{alphabeta}, partial_{beta}$$ and hence
$$square = partial_alphapartial^alpha = partial_alpha eta^{alphabeta}partial_beta = partial^beta partial_beta quad .$$
We have used the Minkowski space metric $(+1,-1,-1,-1)$.

Agnius Vasiliauskas · Answer

If operators commute, namely if commutator :
$$ [ hat {x},hat {y}] = hat x hat y - hat y hat x = 0$$, then there's no difference in order of operators applied, as in this case, because we know Clairaut's theorem :
$$ {frac {partial }{partial x_{i}}}left({frac {partial f}{partial x_{j}}}right) = {frac {partial }{partial x_{j}}}left({frac {partial f}{partial x_{i}}}right) $$
So these two expressions :
$$ square = partial _alpha partial^alpha = partial ^alpha partial_alpha $$
are equivalent.

Are there differences in notation for the d'Alembert operator?

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