ok, so a way to recast the ambient coordinates thing is as follows: Let $S\subset M$ be a smooth submanifold, $\iota\colon S\rightarrow M$ the inclusion and $\Phi\colon S\rightarrow\mathbb{R}$ a smooth function. Assume it possesses a smooth extension $\tilde{\Phi}\colon M\rightarrow\mathbb{R}$. Fix $p\in S$ and a chart $(U,x^1,...,x^n)$ of $M$ (not necessarily adapted) around $p$.
Then $d\tilde{\Phi}=\sum_ia_idx^i$ on $U$ for some smooth functions $a_i$. It follows that $d\Phi=d\iota^{\ast}\tilde{\Phi}=\iota^{\ast}d\Phi=\iota^{\ast}(\sum_ia_idx^i)=\sum_i\iota^{\ast}a_i\iota^{\ast}dx^i=\sum_…

Now if $M\subseteq\mathbb{R}^N$ is a smooth manifold of dimension $n$, we have the tangent bundle $TM=\{(x,v)\in M\times\mathbb{R}^N\vert v\in T_xM\}\subseteq\mathbb{R}^{2N}$. Let $\phi\colon U\rightarrow\mathbb{R}^n$ be a chart for $U\subseteq M$ open.
For $y\in U$, $d\varphi_y\colon T_yU\rightarrow T_{\varphi(y)}\varphi(U)=\mathbb{R}^n$ is an isomorphism. Let $p\colon TM\rightarrow M$ be the canonical projection. Let $\psi\colon p^{-1}(U)\rightarrow U\times\mathbb{R}^n,\,(y,v)\mapsto(y,d\varphi_y(v))$. This ought to be a chart for $TM$.