Christoffel symbol

English

Etymology

Named for Elwin Bruno Christoffel (1829–1900).

Noun

Christoffel symbol (plural Christoffel symbols)

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

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

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