Noun
Christoffel symbol (plural Christoffel symbols)
(differential geometry) For a surface with parametrization
x
→
(
u
,
v
)
{\displaystyle {\vec {x}}(u,v)}
, and letting
i
,
j
,
k
∈
{
u
,
v
}
{\displaystyle i,j,k\in \{u,v\}}
, the Christoffel symbol
Γ
i
j
k
{\displaystyle \Gamma _{ij}^{k}}
is the component of the second derivative
x
→
i
j
{\displaystyle {\vec {x}}_{ij}}
in the direction of the first derivative
x
→
k
{\displaystyle {\vec {x}}_{k}}
, and it encodes information about the surface's curvature. Thus,
[
x
→
u
u
x
→
u
v
x
→
v
v
]
=
[
Γ
u
u
u
Γ
u
u
v
l
Γ
u
v
u
Γ
u
v
v
m
Γ
v
v
u
Γ
v
v
v
n
]
[
x
→
u
x
→
v
n
→
]
{\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
{
l
,
m
,
n
}
{\displaystyle \{l,m,n\}}
is the second fundamental form and
n
→
{\displaystyle {\vec {n}}}
is the surface normal.