Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary of dimension $n$, and let $(U,\varphi)$ and $(V,\psi)$ be charts in the chosen smooth atlas. We say that these charts are positively compatible if either

or else the following condition holds.

For every point

consider any local smooth extension

of the transition map $\psi\circ\varphi^{-1}$ at $x$ that is furnished by the smooth-compatibility condition from Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space. Then

where the Jacobian determinant is the one from Jacobian Determinant of a Differentiable Map Between Euclidean Open Sets.

This condition is independent of the chosen local extension, because $x$ lies in the interior of the half-space chart image and any two such extensions agree on an open neighborhood of $x$ in $\mathbb{R}^n$.