Let \(U\subset \mathbb {R}^n\) be open and \(f:U\to \mathbb {R}^n\) a smooth immersion. Then for each \(x\in U\) we have that \(T_xU\) is canonically isomorphic to \(\mathbb {R}^n\) so
we can define a smooth map \(F:U\times \mathbb {R}^n\to \mathbb {R}^n\times \mathbb {R}^n\) by \[ F(x,v)=(f(x),f_*(x)(v)). \] Show that the derivative \(F_*(x,v):T_{(x,v)}(U\times \mathbb {R}^n)\to T_{F(x,v)}(\mathbb {R}^n\times \mathbb {R}^n)\) has positive determinant as a linear map from \(\mathbb {R}^n\times \mathbb {R}^n= \mathbb {R}^{2n}\) to itself. Use this to show that for any smooth manifold \(M\) its tangent bundle \(TM\) is orientable.