Behavior near the regular point of a smooth vector field

Question

I'm reading Warner's Foundations of Differentiable Manifolds and Lie groups and come up with some trouble in understanding the proof of the theorem about the regular point of a smooth vector field. Here are the statement and the proof:

I barely understand the last part of the proof. I don't understand why we can know $d\sigma$ is non-singular by just computing the values of the differential at each tangent vectors $\frac{\partial}{\partial r_i}|_0$ in $T_0\mathbb{R}^n$. I mean we don't really know what $\frac{\partial}{\partial r_i}|_0$ is for $i\geq 2$, do we? Also, I don't understand why, by choosing the new coordinate system, we can obtain $d\sigma \left( \frac{\partial}{\partial r_1}|_\left( t,a_2,\cdots,a_n\right)\right)=X_{\sigma \left( t,a_2,\cdots,a_n\right)}$ and thus we obtain the conclusion.

Any help or comment are hugely appreciated!!

Answer 1

For your first question:

I don't understand why we can know $d \sigma$ is non-singular by just computing the values of the differential at each tangent vectors $\frac{\partial}{\partial r_i}$ in $T_0\Bbb{R}^n$

The differential $d \sigma$ is a linear map. It is just a basic fact from linear algebra that if the images of basis vectors under a linear map are linearly independent, then the map is non-singular.

Second:

I mean we don't really know what $\left. \frac{\partial}{\partial r_i} \right|_0$ is for $i \geq 2$, do we?

I assume $r_i$ are the standard coordinates on $\Bbb{R}^n$, so that the $\frac{\partial}{\partial r_i}$ are just the standard coordinate vector fields on $\Bbb{R}^n$.

Lastly:

Also, I don't understand why, by choosing the new coordinate system, we can obtain $d \sigma \left( \frac{\partial}{\partial r_1} \right) = X_{σ(t,a_2,⋯,a_n)}$ and thus we obtain the conclusion.

Maybe you will have to look in your book at what Corollary (a) of 1.30 says, but the proof says that this implies that $\sigma$ is a coordinate map (in other words, $t,a_2,\dots,a_n$ are local coordinates near $m$). It is noted earlier in the proof that $d \sigma \left( \left. \frac{\partial}{\partial r_1} \right|_0 \right) = X_m$. To see why this is true not just at zero, think back to what the parameter $t$ represents:

You take the surface defined by the equation $y_1=0$, and what $\sigma$ does is "flow" the surface for time $t$ along the vector field $X$. Since $t$ is the first coordinate (corresponding to $r_1$), $d \sigma \left( \frac{\partial}{\partial r_1} \right)$ is the vector field pointing in the $t$-direction. In other words, it points along the flow lines of $X$.