Notation of multivariable derivatives I reading a text where quadratic terms of a function $L : \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^2$ are expanded as
$$
L(x,\alpha)=\frac{1}{2}L_{20,0} \, x_0^2 + L_{20,1} \, x_0 x_1 + \frac{1}{2}L_{20,2} \, x_1^2 + \frac{1}{2}L_{02,2} \, \alpha_2^2 + L_{12,0} \, \alpha_1 x_0 + L_{12,1} \alpha_1 x_1 + L_{21,0} \, \alpha_2 x_0 + L_{21,1} \, \alpha_2 x_1
$$
where $x=(x_0,x_1)$ and $\alpha=(\alpha_1,\alpha_2)$.
The problem is that I'm not familiar with this index/comma notation of the derivative of a function. 
 A: Assuming that your mapping $$L : \mathbb{R^2}\times\mathbb{R^2} \to \mathbb{R^2}$$ 
maps  $$((x_{11}, x_{12}),(x_{21}, x_{22})) \mapsto (y_1,y_2)$$
the first order derivative should be something like 
$$ \left( \begin{array}{ccc}
\frac{\partial y_1}{\partial x_{11} } & \frac{\partial y_2}{\partial x_{11} }  \\
\frac{\partial y_1}{\partial x_{12} } & \frac{\partial y_2}{\partial x_{12} }  \\
\frac{\partial y_1}{\partial x_{21} } & \frac{\partial y_1}{\partial x_{21} }  \\
\frac{\partial y_1}{\partial x_{22} } & \frac{\partial y_2}{\partial x_{22} } \end{array} \right)$$
Now I'm guessing that the terms $L_{ijk}$ corresponds to some $\dfrac{\partial y_n}{\partial x_{ml} }$. 
I might be dead wrong and it might be that some of the indices should be interpreted as order of the derivative (e.g. a index 2 noting second derivative) or something entirely different.
It's rather messy, especially due to the choice of indexing some of the objects with $\{0,1\}$ and others with $\{1,2\}$, but it think that one should be able to make sense out of it.
The fact that the expression includes the undefined vector $\beta$ does not subtract from the confusion.
Hope this helps...
