$v$ is a critical point of $\operatorname{exp}_{p}$ if and only if $q$ is conjugate to $p$ along $\gamma$ I'm reading John M. Lee 's Introduction to Riemannian manifolds p.299, prop.10.20


I'm trying to understand the underlined statements.
Q.1. First, can anyone explain the equality (upto $T_{v}(T_{p}M) \cong T_{p}M$)  more detail?
Q.2. Second, Why the $J(t)$ is nontrivial for $t$ other than $0,1$ ?
Can any one help?
 A: Q.1 This is as Arctic Char explains in the comments. For any vector $v\in V$, the vector $w\in V$ determines an element of $T_vV$, i.e. a derivation of $C_v^{\infty}(V)$, by mapping $[f]_v\mapsto\frac{\partial}{\partial s}\Big\vert_{s=0}f(v+sw)$ (here, $[f]_v$ denotes a germ at $v$ represented by $f$). This gives you a canonical isomorphism $V\rightarrow T_vV$. For $V=\mathbb{R}^n$, this is proven at the beginning of any smooth manifold text, and the general case follows immediately since the smooth structure one puts on a finite-dimensional vector space $V$ is the one pulled back from $\mathbb{R}^n$ by an arbitrary linear isomorphism.
Q.2 The Jacobi field being non-trivial does not mean that $J(t)\neq0$ for all $t\in(0,1)$. Instead, it means that $J\neq0$, i.e. that not $J(t)=0$ for all $t\in[0,1]$. Note that the zero vector field $0$ is a Jacobi field. Since a Jacobi field is uniquely determined by its value and covariant derivative at any point, we see that $J=0$ if and only if $D_tJ(0)=0$. But Lemma 10.9 calculates $D_tJ(0)=w$ and $w\neq0$ by assumption. This is the crucial point where the non-triviality of $w$ is required.
