I'm looking at "Foundations of Differentiable Manifolds" by Frank Warner, and have a question about one of the basic definitions at the beginning of the book. He writes:
A coordinate system $(U,\phi)$ is called cubic if $\phi(U)$ is an open cube about the origin in $\mathbb{R}^{d}$. A coordinate system is said to be entered at $m\in U$ if $\phi(m)=0$.
Can anyone please explain in pedestrian (i.e. suitable for an undergraduate) terms, what exactly "cubic" means? Could you explicitly give an example? And/or alternative, more concrete definition?
Also, the same question applies to "centered".
Thank you for clarification.
So I would take this as $\phi(U)= (a_1,b_1) \times... \times (a_n,b_n)$, where $a_i<0<b_i$ for each $i$ and $b_1-a_1=b_2-a_2=...=b_n-a_n$. So it looks like a cube in $\mathbb{R}^n$ containing the origin.