... explore this.)UUMϕ( )RnϕA C∞atlas is an indexed collection of charts {(Uα, φα)} which satisfies two conditions:1. The union of the Uαis equal to M; that is, the Uαcover M.2. The ... bgdfdη=adfdλ+ bdfdηg +adgdλ+ bdgdηf . (2.8)As we had hoped, the product rule is satisfied, and the set of directional derivatives istherefore a vector space.2 MANIFOLDS 34xxxone-to-one,...