Christoffel symbols of an affine connection

For an affine connection as in Affine connection on a smooth manifold and coordinates $(x^1,\dots ,x^n)$, its Christoffel symbols in this chart are the unique smooth functions ${\mathrm{\Gamma }}^k{}_{ij}$ such that ${\mathrm{\nabla }}_{{\mathrm{\partial }}_i}{\mathrm{\partial }}_j={\sum }_k{\mathrm{\Gamma }}^k{}_{ij}{\mathrm{\partial }}_k\mathrm{.}$ The first lower index $i$ is the differentiating direction and the second lower index $j$ labels the differentiated vector field. In the coordinate frame, the one-forms from Connection one form in a local frame are ${\omega }^k{}_j={\sum }_i{\mathrm{\Gamma }}^k{}_{ij}d{x}^i$. A different chart generally has different symbols. This definition asserts neither symmetry in $i,j$ nor tensorial transformation.

For dimension one there is one smooth coefficient ${\mathrm{\Gamma }}^1{}_{11}$; in dimension zero there are no coefficients. All statements are local, and at a boundary chart the derivatives are the smooth up-to-boundary coordinate derivatives.