Hi. For a diffeomorphism between 2 manifolds Φ : M → N with a tangent vector v in M I have the following equation for the push-forward of v ( Φ*v)f = v( Φ*f) where Φ* is the pull-back. I understand this equation but i have also come across the following equation for the push-forward (Φ*v)f = (Φ-1)*v (Φ*f ) . Surely these 2 equations are not the same. I'm confused.