i need help, i need to prove these two proposition but i did not have any idea about how should i start for proof:
1)let $M$ be a smooth manifold and $N\subset M$ is sub-manifold of $M$. prove that
$$TN\subset TM$$
2)If $M$ be a smooth manifold and oriented then $TM$ is oriented too.
i really have problem to start solving manifold problems and didn't have any idea about proof of those questions,please help me for solution of them.thank you all.
(Assuming you defines $T$ in terms of equivalent classes of smooth curves.) For $x\in N$, we have a canonical linear map $T_xN\to T_xM$, which is induced by the fact that every curve $\gamma\colon (-\epsilon,\epsilon)\to N$ with $\gamma(0)=x$ can also be viewed as $(-\epsilon,\epsilon)\to M$ (and clearly, equivalent curves in the sense of the definition of $T_xN$ map to equivalent curves in the sense of the definition of $T_xM$). Remains to show that this canonical linear map is injective, thereby allowing us to view $T_xN$ as subspace of $T_xM$. From this statement for stalks, transfer first to small neighbourhoods (i.e., consider small open $U\subset M$ such that both $U$ and $U\cap N$ are contractible), and ultimately to the global tangential bundle.
Given an orientable atlas of $M$ (i.e., such that the Jacobian of transition maps between charts has positive determinant), you can readily construct such an atlas for $TM$.