I am trying to prove the following:
Lemma. Let $S$ be an embedded smooth submanifold of a smooth manifold $M$, let $f\in C^\infty(M)$ be such that $f|_S=0$, and pick $p\in S$. There exists adapted local coordinates $x^1,\dots,x^n$ around $p$ of $M$ with respect to $S$, with domain $U\subset M$ (i.e., $U\cap S=\{x^{s+1}=0,\dots,x^m=0\}$), and smooth functions $g_{s+1},\dots,g_m\in C^\infty(U)$ such that $$ f=\sum_{i=s+1}^m g_ix^i $$ at $U$.
How can one proceed?
$\def\d{\mathrm{d}}$Let $(U,\varphi)$ be an adapted chart of $M$ to $S$, centered at $p$, and such that $\varphi(U)$ is convex. Then in terms of the local coordinates we have \begin{align*} f(x)&=\sum_{i=s+1}^mf(x_1,\dots,x_j,0,\dots,0)-f(x_1,\dots,x_{j-1},0,\dots,0)\\ &=\sum_{i=s+1}^m\int_0^1\frac{\d}{\d t}f(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m\int_0^1x_i\partial_if(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m x_i\int_0^1\partial_if(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m x_ig_i, \end{align*} where $$ g_i=\int_0^1\partial_if(x_1,\dots,tx_i,0\dots,0)\d t,\qquad i=s+1,\dots,m. \quad\square $$
