Given a simple surface $\vec x(u,v)$ with following first fundamental form coefficients: $g (u,v)=\begin{bmatrix} 1 & 0 \\ 0 & g_{uv}(u,v) \end{bmatrix}$, my professor has noted the following: $<\frac{\partial \vec x^2}{\partial u^2}, \frac{\partial \vec x}{\partial v}>=\Gamma_{uu|v}=0$.
I understand that this is zero but I don't understand the first equation. Is this simply the definition of the Christoffel symbols of the first kind?
A definition (not the unique) of the Chrisoffel symbols is the following:
Given a regular surface $\vec{x}(u,v)$ in $\mathbb{R}^3$, if we set
$$
\vec{x}_u=\frac{\partial\vec{x}}{\partial u}\\
\vec{x}_v=\frac{\partial\vec{x}}{\partial v}
$$
and
$$
\vec{n}=\frac{\vec{x}_u\times\vec{x}_v}{|\vec{x}_u\times\vec{x}_v|}
$$
then the system of vectors $B=\left\{\vec{x}_u,\vec{x}_v,\vec{n}\right\}$ is a basis.
The Christoffel symbols can be defined as the coefficients of the expansion, in such a basis, of the second derivatives of $\vec{x}$, i.e.
$$
\vec{x}_{uu}=
\frac{\partial^2 \vec{x}}{\partial u\partial u}=
\Gamma_{uu}^u\vec{x}_u+\Gamma_{uu}^v\vec{x}_v+L\vec{n}\\
\vec{x}_{uv}=
\frac{\partial^2 \vec{x}}{\partial u\partial v}=
\Gamma_{uv}^u\vec{x}_u+\Gamma_{uv}^v\vec{x}_v+M\vec{n}\\
\vec{x}_{vv}=
\frac{\partial^2 \vec{x}}{\partial v\partial v}=
\Gamma_{vv}^u\vec{x}_u+\Gamma_{vv}^v\vec{x}_v+N\vec{n}
$$
Ref.: Pressley, Elementary Differential Geometry, p.172.