Editing Orientability (section)

===Manifolds with boundary===

If ''M'' is a manifold with boundary, then an orientation of ''M'' is defined to be an orientation of its interior.  Such an orientation induces an orientation of ∂''M''.  Indeed, suppose that an orientation of ''M'' is fixed.  Let {{math|''U'' → '''R'''<sup>''n''</sup><sub>+</sub>}} be a chart at a boundary point of ''M'' which, when restricted to the interior of ''M'', is in the chosen oriented atlas.  The restriction of this chart to ∂''M'' is a chart of ∂''M''.  Such charts form an oriented atlas for ∂''M''.

When ''M'' is smooth, at each point ''p'' of ∂''M'', the restriction of the tangent bundle of ''M'' to ∂''M'' is isomorphic to {{math|''T''<sub>''p''</sub>∂''M'' ⊕ '''R'''}}, where the factor of '''R''' is described by the inward pointing normal vector.  The orientation of ''T''<sub>''p''</sub>∂''M'' is defined by the condition that a basis of ''T''<sub>''p''</sub>∂''M'' is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis of ''T''<sub>''p''</sub>''M''.