Let $M$ be a smooth manifold and $N_1,N_2\subseteq M$ two submanifolds intersecting transversely, i.e. $$T_pN_1+T_pN_2=T_pM$$ for all $p\in N_1\cap N_2$. It is a standard result that $N_1\cap N_2$ is a smooth submanifold of $M$.
Let $p\in N_1\cap N_2$. Can we find a local chart $\varphi:U\to \mathbb{R}^m$ for $M$ around $p$ such that $$ \begin{align*} \varphi(N_1\cap U) &= \mathbb{R}^{n_1}\times\{0\} \\ \varphi(N_2\cap U) &= \{0\}\times\mathbb{R}^{n_2} \end{align*} $$ and hence $\varphi(N_1\cap N_2\cap U)=\{0\}\times\mathbb{R}^{m-n_1-n_2}\times\{0\}$?