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Definition 2025

Christoffel_symbol

Christoffel symbol

English

Noun

Christoffel symbol ‎(plural Christoffel symbols)

(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.