Let \(F\colon M\to N\) denote a smooth map between manifolds . The global differential or global tangent map, denoted by \(dF\colon TM\to TN\), is just the map whose restriction to each tangent space \(T_pM\subset TM\) is the differential \(dF_p\).
Remarks
Suppose \(F\colon M\to N\) and \(G\colon N\to P\) are smooth maps.
- \(d(G\circ F) = dG\circ dF\).
- \(d(\id_M )= \id_{TM} \).
- If \(F\) is a diffeomorphism , then \(dF\colon TM\to TN\) is also a diffeomorphism, and \((dF)^{-1}=d(F^{-1})\).