When a smooth curve in a manifold is an immersion?

Question

I need help with the example 4.2 (b) page 78, in the book Introduction to Smooth Manifolds by John Lee. The example says that a smooth curve $\gamma:J\to M$ in a smooth manifold $M$ is smooth immersion if and only if $\gamma'(t)$ is not zero, for all $t \in J$. I use the definitions but I do not have the perfect conclusion for each direction. It might be obvious but I have been stuck for days. I really need assistance because it's self study. I would appreciate any help.

Answer 1

You need to show that for each $t$, the differential $d\gamma_t$ is injective iff the velocity $\gamma'(t)\neq0$.

And $d\gamma_t$ is injective iff the Jacobian matrix of $\gamma$ has rank $1$.

In coordinates in a smooth chart, $$\gamma'(t_0)=\frac{d\gamma^i}{dt}(t_0)\frac{\partial}{\partial x^i}\Big|_{\gamma(t_0)}$$ and the Jacobian matrix of $d\gamma_{t_0}$ is $\begin{pmatrix}\frac{d \gamma^1}{dt}(t_0)\\\vdots\\\frac{d \gamma^n}{dt}(t_0)\end{pmatrix}$ which has rank $1$ iff $\frac{d\gamma^i}{dt}(t_0)\neq 0$ for some $i$ iff $\gamma'(t_0)\neq 0$.