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:
“I dare say I am compelled, unconsciously compelled, now to write volume after volume, as in past years I was compelled to go to sea, voyage after voyage. Leaves must follow upon each other as leagues used to follow in the days gone by, on and on to the appointed end, which, being Truth itself, is One—one for all men and for all occupations.”
—Joseph Conrad (1857–1924)
“That is what the highest criticism really is, the record of one’s own soul. It is more fascinating than history, as it is concerned simply with oneself. It is more delightful than philosophy, as its subject is concrete and not abstract, real and not vague. It is the only civilised form of autobiography.”
—Oscar Wilde (1854–1900)