Editing Real coordinate space (section)

=== Orientation ===
The fact that [[real numbers]], unlike many other [[field (mathematics)|fields]], constitute an [[ordered field]] yields an [[orientation (vector space)|orientation structure]] on {{math|'''R'''<sup>''n''</sup>}}. Any [[rank (matrix theory)|full-rank]] linear map of {{math|'''R'''<sup>''n''</sup>}} to itself either preserves or reverses orientation of the space depending on the [[sign (mathematics)|sign]] of the [[determinant]] of its matrix. If one [[permutation|permutes]] coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the [[parity of a permutation|parity of the permutation]].

[[Diffeomorphism]]s of {{math|'''R'''<sup>''n''</sup>}} or [[domain (mathematical analysis)|domains in it]], by their virtue to avoid zero [[Jacobian matrix and determinant|Jacobian]], are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of [[differential form]]s, whose applications include [[electrodynamics]].

Another manifestation of this structure is that the [[point reflection]] in {{math|'''R'''<sup>''n''</sup>}} has different properties depending on [[even and odd numbers|evenness of {{mvar|n}}]]. For even {{mvar|n}} it preserves orientation, while for odd {{mvar|n}} it is reversed (see also [[improper rotation]]).