Editing Curvilinear coordinates (section)

===Christoffel symbols===

;[[Christoffel symbols]] of the first kind <math>\Gamma_{kij}</math>:

:<math>
\mathbf{b}_{i,j} = \frac{\partial \mathbf{b}_i}{\partial q^j} = \mathbf{b}^k \Gamma_{kij} \quad \Rightarrow \quad
\mathbf{b}_k  \cdot \mathbf{b}_{i,j}  = \Gamma_{kij}
</math>

where the comma denotes a [[partial derivative]] (see [[Ricci calculus]]). To express Γ<sub>''kij''</sub> in terms of ''g<sub>ij</sub>'',

:<math>
\begin{align}
g_{ij,k} & = (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k} = \mathbf{b}_{i,k}\cdot\mathbf{b}_j + \mathbf{b}_i\cdot\mathbf{b}_{j,k}
= \Gamma_{jik} + \Gamma_{ijk}\\
g_{ik,j} & = (\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} = \mathbf{b}_{i,j}\cdot\mathbf{b}_k + \mathbf{b}_i\cdot\mathbf{b}_{k,j}
= \Gamma_{kij} + \Gamma_{ikj}\\
g_{jk,i} & = (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} = \mathbf{b}_{j,i}\cdot\mathbf{b}_k + \mathbf{b}_j\cdot\mathbf{b}_{k,i}
= \Gamma_{kji} + \Gamma_{jki}
\end{align}
</math>

Since
:<math>\mathbf{b}_{i,j} = \mathbf{b}_{j,i}\quad\Rightarrow\quad\Gamma_{kij} = \Gamma_{kji}</math>
using these to rearrange the above relations gives

:<math>\Gamma_{kij} = \frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k}) = \frac{1}{2}[(\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} + (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} - (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k}]
</math>

;[[Christoffel symbols]] of the second kind <math>\Gamma^k{}_{ji}</math>:

:<math>\Gamma^k{}_{ij} = g^{kl}\Gamma_{lij} = \Gamma^k{}_{ji},\quad \cfrac{\partial \mathbf{b}_i}{\partial q^j} = \mathbf{b}_k \Gamma^k{}_{ij} </math>

This implies that
:<math> \Gamma^k{}_{ij} = \cfrac{\partial \mathbf{b}_i}{\partial q^j}\cdot\mathbf{b}^k = -\mathbf{b}_i\cdot\cfrac{\partial \mathbf{b}^k}{\partial q^j}\quad </math> since <math> \quad\cfrac{\partial}{\partial q^j}(\mathbf{b}_i\cdot\mathbf{b}^k)=0</math>.

Other relations that follow are
:<math>
\cfrac{\partial \mathbf{b}^i}{\partial q^j} = -\Gamma^i{}_{jk}\mathbf{b}^k,\quad
\boldsymbol{\nabla}\mathbf{b}_i = \Gamma^k{}_{ij}\mathbf{b}_k\otimes\mathbf{b}^j,\quad
\boldsymbol{\nabla}\mathbf{b}^i = -\Gamma^i{}_{jk}\mathbf{b}^k\otimes\mathbf{b}^j
</math>