Answer to Question #304454 in Differential Geometry | Topology for Rkjgsv

Question

Let M and N be smooth manifolds and let πM : M × N → M and πN : M × N → N

be the projection maps. For any (p, q) ∈ M × N show that the map

Π : Tp(M × N) → TpM × TpN,

defined by

Π(v) = (D(πM)p(v), D(πN )q(v))

is an isomorphism