Assume we work in a strictly convex domain $\Omega$ with $C^1$ boundary, say in $\mathbb{R}^3$. Given two points at the boundary $x$ and $y$, I first wanted to compute the corresponding unit vector linking $x$ to $y$. It is given by $$ u = \frac{x - y}{\|x - y\|}.$$
On the other hand, because of the convexity assumption, we will always have that the condition $u \cdot n_x < 0$ is true for any vector of this kind (where $n_x$ is the outward normal vector to $\Omega$ at the point $x$).
Now I want to compute the Jacobian of the map $$ \psi_x: \partial \Omega \to \{u \in \mathbb{S}^2, u \cdot n_x < 0 \}, $$ $$ y \to \psi_x(y) = \frac{x - y}{\|x - y\|}. $$
Note that this is indeed a $C^1$-diffeomorphism: we start for a point in an hypersurface and send it into another hypersurface, there are no dimensional issues. I am having trouble computing this Jacobian matrix. I am not confortable at all with this kind of diffeomorphism between manifolds with boundary. Any clue on how to get started ?