Proof of Cartan's theorem in Do Carmo's book I would like to know why it is necessary to prove that $\widetilde{J}(l) = df_q(v) = df_q(J(l))$ to prove the Cartan's theorem below.
Thanks in advance!


I realized that the equality in the post is necessary after observe that $J(l) = v$, $\widetilde{J}(l) = df_q(v)$ and $|\widetilde{J}(l)| = |J(l)|$, then $|df_q(v)| = |v|$.
