Christoffel symbols of a connection

Definition

Let $M$ be a differential manifold, $E$ a vector bundle over $M$, and $\mathrm{\nabla }$ a connection on $E$. The Christoffel symbols of $E$ are a way to describe this connection using a set of numbers, that vary from point to point.

To describe the Christoffel symbols, we choose a local coordinate chart Fill this in later