# Constant rank theorem: intuition?

Let $f: \mathbb R^n \to \mathbb R^m$ be smooth and let $x_0 \in \mathbb R^n$ be such that $\operatorname{rank}{(J_f(x_0))} = k$. Then there exists a neighboudhood of $x_0$ and diffeomorphisms $\phi, \psi$ such that

$$\phi \circ f \circ \psi = (x_1, \dots, x_n ) \mapsto (x_1, \dots, x_k, 0 \dots, 0)$$

on this neighbourhood.

This is the constant rank theorem.

It seems to me that this is saying that any smooth map can be written as a projection onto some of its coordinates on some neighbourhood.

Is this really what this is saying? Please could someone help me get the intuition behind this theorem?

• As Areaman's answer states this is an incorrect statement of the theorem, for example for $f:\Bbb R \to \Bbb R$ with $f(x)=x^2$ restricted to any neighborhood of zero does not look like the zero map locally. – PVAL-inactive Aug 23 '15 at 2:24