In continuation to a previous question, consider that I want to extend the calculation, and also calculate and numerically evaluate the correlation between the principal curvatures and their derivatives w.r.t. the principal directions. So, for example, I would like to calculate the correlation between $\kappa_1$ and $\kappa_{1,1}$, where $\kappa_{1,1}$ is the derivative of $\kappa_1$ w.r.t. the first principal direction. So, consider that I follow the same structure as given in the answer to the question referenced above, and use a quadratic Monge patch: $$ h\left(u,v\right) = \frac{X}{2}u^2 + \frac{Y}{2}v^2$$ Where $X$ and $Y$ are two uniformly distributed random variables.
In this case, the second fundamental form matrix is a diagonal matrix, so if I understand correctly, at $\left(0,0\right)$ the principal directions should be aligned with the coordinate directions of the parametrization plane. Does it mean that in this case, the derivatives of $h$ w.r.t. $u$ and $v$ are equivalent to the derivatives of $h$ w.r.t. the principal directions $d_1$ and $d_2$?
If that's true, then since $\kappa_1 = H + \sqrt{H^2 - K}$, and $H$ and $K$ can be expressed as functions of $h$ and its derivatives, then I can differentiate $\kappa_1$ w.r.t. to $u$ in order to find $\kappa_{1,1}$, and finally, evaluate $\kappa_{1,1}$ at $\left(0,0\right)$ to express it as a function $X$ and $Y$.