Christoffel symbol

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 ${\\displaystyle {\\vec {x}}(u,v)}$, 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.