Let be a local coordinate system on the base manifold with and let .
\begin{cases}\psi:W \to \varphi(U)\times \mathbf{R}^N \\ \psi \left (v^k e_k|_x \right ) := \left (x^1,\ldots,x^n,v^1,\ldots,v^N \right )\end{cases} .