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Here is what I'm experiencing.

A text introduces manifolds. Then, manifolds with boundary is introduced. After all, manifolds with corners is introduced.

First of all, I checked whether Implicit function theorem (IFT) and Clairaut's theorem and etc for open subsets of $\mathbb{R}^n$. After that, when manifolds with boundary is introduced, I checked again whether IFT and Clairaut's theorem and etc for open subsets of closed upper half-space. After that, when manifolds with corners is introduced, I chekced again whether IFT and Clairaut's theorem and etc for open subsets of $H_i^n:=\{x\in\mathbb{R}^n : x_{n+1-i},...,x_n \geq 0 \}$ ($0\leq i \leq n$).

I'm sick and tired of this checking again and again stuff. It would be great if manifolds of corners were first introduced so that I can check multivariable calculus stuff at one shot.

I'm wondering if there is a more standard generalization of manifolds with corners, so that I can check it at one shot.

Moreover, is there a terminology for $H_i^n$? Is it okay to call it "the $(n,i)$ -upper half-space"?

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    $\begingroup$ Sometimes stuff gets generalized. And the generalization gets generalized. And so on. Sometimes, it's generalizations all the way down. zazzle.com/… $\endgroup$ – Lee Mosher Aug 19 '16 at 16:25
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The answer to your question is yes. Manifolds with boundary and manifolds with corners are special cases of subcartesian differential spaces introduced by Aronszajn in 1967. It is surprising how much standard differential geometry can be reproduced in subcartesiam spaces. Reference, Differential Geometry of Singular Spaces and Reduction of Symmetry by J. Sniatycki, Cambridge University Press 2013.

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