Volume Form
Let ω be a form on a n-dimensional real vector space V, ω ∈ Λ2(V). Then ω is non-degenerate if and only if n is even, and ωn/2 = 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 e1∗ ∧ ... ∧ en∗ where e1, e2, ..., en is a basis of V.
For the standard basis defined in the previous section, we have
By reordering, one can write
Authors variously define ωn or (−1)n/2ωn 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 on the symplectic vector space (V, ω).
Read more about this topic: Symplectic Vector Space
Famous quotes containing the words volume and/or form:
“We are too civil to books. For a few golden sentences we will turn over and actually read a volume of four or five hundred pages.”
—Ralph Waldo Emerson (1803–1882)
“It is against Stupidity in every shape and form that we have to wage our eternal battle. But how can we wonder at the want of sense on the part of those who have had no advantages, when we see such plentiful absence of that commodity on the part of those who have had all the advantages?”
—William Booth (1829–1912)