- #1
center o bass
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There are two theorems from multivariable calculus that is very important for manifold theory.
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?
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?