Pointwise orientation sign of a local diffeomorphism

By [L1] and [L2], $d{F}_p$ is an isomorphism. Its determinant map therefore sends the selected source ray to exactly one of the two target rays. Choose local nonzero determinant sections ${\sigma }_M$ and ${\sigma }_N$ in the selected rays. There is a smooth nowhere-zero scalar $\lambda$ such that $\det (dF)({\sigma }_M)=\lambda {\sigma }_N$; its sign is precisely whether the selected rays are preserved or reversed.

Since $\lambda$ is continuous and never zero, its sign is locally constant and therefore constant when $M$ is nonempty and connected. If oriented source and target charts happen to be available, their coordinate determinants may be used for ${\sigma }_M$ and ${\sigma }_N$, and then $\lambda$ is the Jacobian determinant. The determinant-line formulation remains valid at one-dimensional boundary points and in dimension zero, where such chart frames need not encode every selected ray.