I have some trouble understanding the definition of a regular submanifold. I came across this definition in an introductory course on smooth manifolds, we are using L. W. Tu's book. The definition given in the textbook is the following:

A subset $S$ of a manifold $N$ with $dim(N)=n$ is a regular submanifold of dimension $k$ if for every $p\in S$ there is a chart $(U,\phi)=(U, x^1,\dots, x^n)$ of p in the maximal atlas of $N$ such that $U\cap S=\{q\in U: x^{k+1}(q)=0, \dots, x^{n}(q)=0\}$

My question is this: why do we demand that the LAST $n-k$ coordinates vanish? Is this enough to prove that every set with a proportional property, such as "the first $n-k$ coordinates vanish", is a regular submanifold?

If not, why not? In my intuition it would be more natural to define a submanifold as a subset of a manifold that "needs less information to be described", meaning that, in the inherited property of being locally euclidean, in every chart, some (not specifying which) coordinates wouldn't play a role.

My only idea was that maybe a map that interchanges coordinates is smooth and the definition above is enough, but I can't work it out.

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    $\begingroup$ I think an interchange of coordinates would do no harm, since the map taking one set of coordinates to another is a homeomorphism and we can probably achieve another set of coordinates being set to zero by using a composition. The order of the coordinates therefore should not be significant. $\endgroup$ – астон вілла олоф мэллбэрг Feb 6 '18 at 0:36

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