Suppose $f_1,\dots, f_d$ are a set of real-valued functions on a smooth manifold $M$. Let $N$ be the zero locus of the $f_i$. Suppose the $\textrm{d}f_i$ span a subspace of the cotangent space of $M$ of dimension $d'
I've applied the inverse function theorem to prove that wlog the $f_1,\dots,f_d'$ are local coordinates around any given $p$ in $M$. I'd now like to restrict my diffeomorphism to $N$, but I'm worried that the extra $f_i$ for $i > d'$ will yield some subtleties. In particular my lecturer suggested that I needed to check that the other $f_i$ were constant on all curves through $p$. Why is this, and how does it help? I'm afraid I haven't quite got the intuition for this yet, so any comments would be gratefully appreciated!