Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the differential $ \phi_{ \ast,q}: T_qM \to T_{f(q)}\mathbb{R} \simeq \mathbb{R}^n $ carries the orientation of $T_qM$ to the stander orientation of $\mathbb{R}^n $ in the following sense: $(\phi_{ \ast}X_{1,q},..., \phi_{ \ast}X_{n,q}) \sim (\frac{\partial}{\partial r^1},..., \frac{\partial}{\partial r^n})$
How can i prove this, this exercise is from Introduction to Manifolds Tu,Thanks for your help, I try but really I stuck in this problem.