Need help following through the matrix calculus in this paper I've been making my way through this paper:
Three-dimensional surface curvature estimation using quadric surface patches
Specifically appendix 1 on pages 12-13, the part at the top of page 13:
On differentiating with respect to $u_{\gamma \delta}$:
\begin{equation}
\sum_{x,y,z \in N} x_{\gamma} x_{\delta} \sum_{\alpha\leq\beta} x_{\alpha}u_{\alpha\beta}x_{\beta} - \lambda u_{\gamma\delta} = 0
\end{equation}
How does differentiating by $u_{\gamma \delta}$ yield $x_{\gamma} x_{\delta}$?
Afterwards, why is $\sum_{x,y,z \in N} x_{\gamma} x_{\delta} x_{\alpha} x_{\beta}$ a 10x10 matrix?
This might be some notation confusion, still, I would appreciate links and explanation of the theory and identities the author is using. 
Thanks in advance!
 A: The function they are taking the derivative of is
$$
f = \sum_{x,y,z\in \mathbb{N}}\left(\sum_{\alpha\leq \beta} x_\alpha u_{\alpha \beta}x_\beta\right)^2 + \lambda \left(1 - \sum_{\alpha \leq \beta}u_{\alpha\beta}^2\right) \tag{1}
$$
So the minimum is found by setting $\partial f/\partial u_{\gamma \delta} = 0$. To do that just note that
$$
\frac{\partial u_{\alpha\beta}}{\partial u_{\gamma\delta}} = \delta_{\alpha \gamma}\delta_{\beta\delta} \tag{2}
$$
Taking this into account we get
\begin{eqnarray}
\frac{\partial f}{\partial u_{\gamma \delta}} &=& 2\sum_{x,y,z\in \mathbb{N}} \left(\sum_{\alpha\leq \beta} x_\alpha u_{\alpha \beta}x_\beta\right) \left(\sum_{\alpha\leq \beta}x_\alpha\color{blue}{\frac{\partial u_{\alpha\beta}}{\partial u_{\gamma \delta}}}x_\beta\right) - 2\lambda\sum_{\alpha \leq \beta}u_{\alpha\beta} \color{red}{\frac{\partial u_{\alpha \beta}}{\partial u_{\gamma \delta}}} \\
&\stackrel{(2)}{=}&2\sum_{x,y,z\in \mathbb{N}} \left(\sum_{\alpha\leq \beta} x_\alpha u_{\alpha \beta}x_\beta\right) \left(\sum_{\alpha\leq \beta}x_\alpha\color{blue}{\delta_{\alpha\gamma}\delta_{\beta\delta}}x_\beta\right) - 2\lambda\sum_{\alpha \leq \beta}u_{\alpha\beta} \color{red}{\delta_{\alpha\gamma}\delta_{\beta\delta}} \\
&=&2\sum_{x,y,z\in \mathbb{N}} \left(\sum_{\alpha\leq \beta} x_\alpha u_{\alpha \beta}x_\beta\right) \left(x_\gamma x_\delta\right) - 2\lambda u_{\gamma \delta}
\end{eqnarray}
Rearranging you then have
$$
\sum_{\alpha \leq \beta}\left(\sum_{x,y,z\in\mathbb{N}} x_\gamma x_\delta x_\alpha x_\beta\right) u_{\alpha\beta} = \lambda u_{\gamma\delta} \tag{3}
$$
As for the second question, note that in $u_{\alpha\beta}$ you must satisfy $\alpha\leq \beta$, so that means that $u_{12}$ is allowed, but $u_{21}$ is not. If you count all possible combinations you end up with 10
