$f : X \to Y$ be a map of equidimensional manifolds with regular value $y$, extending to a map $g : Z \to Y$ with $\partial Z = X$, assume that also has $y$ has a regular value. Then $g^{-1}(y)$ is a 1-manifold in $Z$ with boundary $f^{-1}(y)$ in $X$. Orient a connected component $I$ of $g^{-1}(y)$ by orienting the normal bundle, which is precisely pullback of the normal bundle of $y$ (aka $T_y Z$) by $g$.
$dg$ thus sends the normal space $N_p(I; Z)$ for any $p \in I$ by an orientation preserving isomorphism to $T_y Z$. The orientation on $I$ is then given by the vector field $v$ on $I$ suc…