There are two theorems from multivariable calculus that is very important for manifold theory.(adsbygoogle = window.adsbygoogle || []).push({});

The first is the inverse function theorem and the second is the "constant rank theorem". The latter states that

(Constant rank theorem).If ##f : U\subset \mathbb{R}^n \to \mathbb{R}^m## has constant rank ##k## in a neighborhood of a point ##p \in U## , then after a suitable change of coordinates near ##p## in ##U## and ##f(p)## in ##\mathbb{R}^m##, the map ##f## assumes the form ##(x^1,...,x^n)\mapsto (x^1,...,x^k,0,...,0)##.

More precisely, there are a diffeomorphism ##G## of a neighborhood of ##p## in ##U## sending ##p## to the origin in ##\mathbb{R}^n## and a diffeomorphism ##F## of a neighborhood of ##f(p)## in ##\mathbb{R}^m## sending ##f(p)## to the origin in ##\mathbb{R}^m## such that ##(F ◦ f ◦ G)^{−1}(x^1,...,x^n) = (x^1,...,x^k,0,...,0).##

I've gone through the proof of the theorem, but I'm left with little intuition on why it has to be true. Therefore I wonder, do you have any intuitive explanation of why the theorem has to be true?

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# Constant Rank Theorem Intuition

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