In Do Carmo's Differential Geometry of Curves and Surfaces he does the following:
Let $\vec r$ be a parametrization of a surface $S\subset\mathbb{R}^3$ so that $\vec r_u,\vec r_v$ forms a basis for $T_pS$ at each point in the image of $\vec r$. We can then define $N=\frac{\vec r_u\times\vec r_v}{|\vec r_u\times \vec r_v|}\in (T_pS)^\perp$ so that $\{\vec r_u,\vec r_v,N\}$ is a basis for $\mathbb{R}^3$.
We can express $\vec r_{uu},\vec r_{uv}=\vec r_{vu},\vec r_{vv}$ in terms of this basis. We set $$\vec r_{uu}=\Gamma_{11}^1\vec r_u+\Gamma_{11}^2\vec r_v+L_1 N$$ $$\vec r_{uv}=\Gamma_{12}^1\vec r_u+\Gamma_{12}^2\vec r_v+L_2N$$ $$ \vec r_{vv}=\Gamma_{22}^1\vec r_u+\Gamma_{22}^2\vec r_v+L_3N$$
I am failing to recover this definition of the Christoffel symbols from that on an arbitrary Riemannian manifold as $\Gamma_{ij}^k=(\nabla_{E_i}E_j)^k$. There's obviously something simple I am missing. If $\nabla$ is the Euclidean connection on $\mathbb{R}^3$, and $\vec r=(x,y,z)$ then $\nabla_{\vec r_u}\vec r_u=\vec r_u(x_u)+\vec r_u(y_u)+\vec r_u(z_u)$, right? Where is $\vec r_{uu}$ coming in?