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"?