Tangent coordinate transition
= Tangent coordinate transition
{title2=$(x,v)\mapsto(f(x),D f(x)v)$}
A <coordinate chart> $x$ trivializes the <tangent bundle> by expressing $v\in T_pM$ in the coordinate <basis>. For $y=f(x)$, the <chain rule> gives the displayed transition. Smoothness and fibrewise linear invertibility of the <Jacobian matrix> construct the natural smooth <vector bundle> structure. The chain rule also supplies the transition cocycle.