Problem 1. A smooth curve y is a smooth map from an interval / C R to a manifold M. Under the identification 7,/ ~ {} x R, the tangent vector (t, 1), which is just denoted 1, lies in the tangent space THI for each t ( I. For each t ( I, the derivative (t) is defined by y'(t) = Try . 1. Given a chart (U, d = ( , ...;$")) on M such that y(t) E U for all te / C I, define the local version of y by 7% = 0' and the local version of y by Y's (t ) = (boy)'(t). Show that Yo (t ) = dye(t) dyl (t) dyn (t) dt dt dt and dy(t) a Y (t) = dt Op' ly(t) for all te