Let be a differentiable manifold. An orientation of is a family of atlases on such that for all
Proposition (an orientation of a manifold induces an orientation on its boundary):
Let be a differentiable manifold. If is an orientation of , then an orientation of is given by
where is defined as follows:
Proof: The given family of functions defines an atlas on , so that the proposition will be proven once it is demonstrated that the requirement regarding the positivity of the determinant is satisfied.
Indeed, let . Then
Since maps the set to itself, the first row of is zero so long as , except the very first entry. Yet the very last entry must be non-negative so long as , since otherwise the fundamental theorem of calculus and the continuity of the derivative would imply that for a such that was sufficiently small, would be contained within
contrary to the definition of charts that contain a piece of the boundary. Hence, upon carrying out a Leibniz expansion of along the first row, we obtain that the matrix obtained from by removing the first row and the first column has positive determinant. Yet the definitions of the respective partial derivatives as limits show that this matrix is exactly the matrix