Notation of multiple covariant derivatives of a tensor From this wikipedia page the laplacian of a tensor is defined by
$$ \Delta T = g^{ij} \left(\nabla_{\partial_i} \nabla_{\partial_j} T - \nabla_{\nabla_{\partial_i} \partial_j} T\right). $$
I have an issue here with what they mean by $\nabla_{\partial_i} \nabla_{\partial_j} T $. Consider for simplicity the case that $T$ is a $(2,0)$-tensor (input 2 vectors, output a real number). There ware two ways to interpreet this. 


*

*Interpreet $\nabla_{\partial_j} T$ first as the $(3,0)$-tensor $\nabla T$. Then we take the covariant derivative of $\nabla T$ and then insert the vectors $\partial_i$ and $\partial_j$.

*Interpreet $\nabla_{\partial_j} T$ as a $(2,0)$-tensor and take the covariant derivative with respect to $\partial_i$ of this expression.


I know that the first interpretation is independent of the coordinates and the second isn't. When the first interpretation is written out in coordinates, then it equals the formula for $\Delta T$, but interpreted by the second interpretation listed. I feel like the second interpretation is the correct one?
 A: It seems that the consensus is:
The first interpretation is $\nabla^2 T(\partial_i, \partial_j)$. It is usually denoted $\nabla_i\nabla_j T$ (instead of $\nabla_{\partial_i}\nabla_{\partial_j} T$). So 
\begin{align}
\nabla_i\nabla_j T &= \nabla^2 T(\partial_i, \partial_j) \\
&= (\nabla_{\partial_i} \nabla T) (\partial_j) \\
&= \nabla_{\partial_i} (\nabla T(\partial_j)) - \nabla T(\nabla_{\partial_i} \partial_j) \\
&= \nabla_{\partial_i} (\nabla_{\partial_j} T) - \nabla_{\nabla_{\partial_i}\partial_j} T.
\end{align}
On the other hand, if $X$, $Y$ are vector fields, $\nabla_X \nabla_Y T$ means $\nabla_X (\nabla_YT)$ (That is, the second interpretation).
Remark: This choice is quite confusing when you first see it (Indeed, I mixed the two up recently when writing a paper and did not notice for quite some time). Some people refuse to write $\nabla_i \nabla_j T$ and use instead $(\nabla^2 T)_{ij}$, $(\nabla\nabla T)_{ij}$ or $T_{; ij}$. But my feeling is that the majority use $\nabla_i\nabla_j T$. 
