Christoffel symbol

Named for Elwin Bruno Christoffel (1829–1900).

Christoffel symbol (plural Christoffel symbols)

1. (differential geometry) For a surface with parametrization ${\displaystyle {\vec {x}}(u,v)}$ , and letting ${\displaystyle i,j,k\in \{u,v\}}$ , the Christoffel symbol ${\displaystyle \Gamma _{ij}^{k}}$ is the component of the second derivative ${\displaystyle {\vec {x}}_{ij}}$ in the direction of the first derivative ${\displaystyle {\vec {x}}_{k}}$ , and it encodes information about the surface's curvature. Thus,

${\displaystyle {\begin{bmatrix}{\vec {x}}_{uu}\\{\vec {x}}_{uv}\\{\vec {x}}_{vv}\end{bmatrix}}={\begin{bmatrix}\Gamma _{uu}^{u}&\Gamma _{uu}^{v}&l\\\Gamma _{uv}^{u}&\Gamma _{uv}^{v}&m\\\Gamma _{vv}^{u}&\Gamma _{vv}^{v}&n\end{bmatrix}}{\begin{bmatrix}{\vec {x}}_{u}\\{\vec {x}}_{v}\\{\vec {n}}\end{bmatrix}}}$

where ${\displaystyle \{l,m,n\}}$ is the second fundamental form and ${\displaystyle {\vec {n}}}$ is the surface normal.