Christoffel symbol

English

Etymology

Named for Elwin Bruno Christoffel (1829–1900).

Noun

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

$\left[\begin{array}{c}{\overrightarrow{x}}_{uu}\\ {\overrightarrow{x}}_{uv}\\ {\overrightarrow{x}}_{vv}\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{\Gamma }}_{uu}^u& {\mathrm{\Gamma }}_{uu}^v& l\\ {\mathrm{\Gamma }}_{uv}^u& {\mathrm{\Gamma }}_{uv}^v& m\\ {\mathrm{\Gamma }}_{vv}^u& {\mathrm{\Gamma }}_{vv}^v& n\end{array}\right]\left[\begin{array}{c}{\overrightarrow{x}}_u\\ {\overrightarrow{x}}_v\\ \overrightarrow{n}\end{array}\right]$

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