Editing Symplectic vector space (section)

==Volume form==
Let ''ω'' be an [[alternating bilinear form]] on an ''n''-dimensional real vector space ''V'', {{nowrap|''ω'' ∈ Λ<sup>2</sup>(''V'')}}. Then ''ω'' is non-degenerate if and only if ''n'' is even and {{nowrap|1=''ω''<sup>''n''/2</sup> = ''ω'' ∧ ... ∧ ''ω''}} is a [[volume form]].  A volume form on a ''n''-dimensional vector space ''V'' is a non-zero multiple of the ''n''-form {{nowrap|''e''<sub>1</sub><sup>∗</sup> ∧ ... ∧ ''e''<sub>''n''</sub><sup>∗</sup>}} where {{nowrap|''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub>}} is a basis of ''V''.

For the standard basis defined in the previous section, we have

:<math>\omega^n = (-1)^\frac{n}{2} x^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_1 \wedge \dotsb \wedge y^*_n.</math>

By reordering, one can write

:<math>\omega^n = x^*_1 \wedge y^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_n.</math>

Authors variously define ''ω''<sup>''n''</sup> or (−1)<sup>''n''/2</sup>''ω''<sup>''n''</sup> as the '''standard volume form'''. An occasional factor of ''n''! may also appear, depending on whether the definition of the [[alternating product]] contains a factor of ''n''! or not.  The volume form defines an [[orientation (mathematics)|orientation]] on the symplectic vector space {{nowrap|(''V'', ''ω'')}}.