Christoffel symbol

Etymology
Named for Elwin Bruno Christoffel (1829–1900).

Noun

 * 1)  For a surface with parametrization $$\vec x (u,v)$$, and letting $$ i, j, k \in \{u, v\} $$, the Christoffel symbol $$\Gamma_{i j}^k $$ is the component of the second derivative $$ \vec x_{i j} $$ in the direction of the first derivative $$ \vec x_k $$, and it encodes information about the surface's curvature. Thus,
 * $$ \begin{bmatrix} \vec x_{u u} \\ \vec x_{u v} \\ \vec x_{v v} \end{bmatrix} = \begin{bmatrix} \Gamma_{u u}^u & \Gamma_{u u}^v & l \\ \Gamma_{u v}^u & \Gamma_{u v}^v & m \\ \Gamma_{v v}^u & \Gamma_{v v}^v & n \end{bmatrix} \begin{bmatrix} \vec x_u \\ \vec x_v \\ \vec n \end{bmatrix} $$
 * where $$ \{l, m, n\} $$ is the second fundamental form and $$\vec n$$ is the surface normal.