I was reading the proof of Gauss Lemma from do Carmo's book on Riemannian geometry. We need to prove $\langle (d\exp_p)_v(v), (d\exp_p)_v(w) \rangle = \langle v,w\rangle$. He decomposes $w=w_T+w_N$ where (in my understanding) $w_T$ is the "tangential" component and $w_N$ is the "normal" component.
So I have understood the whole proof, but I couldn't understand the motivation or reason behind choosing the parametrised surface $f:A\rightarrow M$ where $M$ is the manifold and $A=\{(t,s)\ : 0\leq t\leq 1, -\epsilon \leq s\leq \epsilon\}$ and $f(t,s)=\exp_{p}\ tv(s)$.
I understood the fact that these are basically a family of curves in $T_p(M)$ and the corresponding image under $f$ for a fixed $s$ gives a family of geodesics. But, I couldn't understand the motivation to consider these family of curves.
Thanks.