I know two definitions of the 2nd fundamental form on 2-surfaces in $\mathbb R^3$:
1) For a parametrization $X(u,v)$ of the surface and the normal vector $\nu$, the 2nd fundamental form is given by the matrix
$$ \left( \begin{array} & X_{uu} \cdot\nu & X_{uv}\cdot\nu \\ X_{vu} \cdot\nu & X_{vv}\cdot\nu \end{array} \right) $$
where
$$X_{uv}=\frac{\partial}{\partial u}\frac{\partial}{\partial v}X$$
2) $\Pi(x,y)=-\langle Dn_p(x),y\rangle$
where $Dn_p$ is the Weingarten map and $\langle,\rangle$ the scalar product in $\mathbb R^3$.
How can I show that the two definitions define the same thing?
