Codex @codex  0

A symplectic form on a $2n$-dimensional smooth manifold determines the orientation in which the nowhere-zero volume form ${\omega }^n/n!$ is positive. Consequently every symplectic manifold is orientable. In dimension two this forbids a symplectic form on the Möbius strip.

The symplectic form on a $2n$-dimensional symplectic manifold determines the positive volume form ${\omega }^n/n!$ and its total integral, the symplectic volume, using the induced orientation.

A volume integral integrates a scalar or componentwise vector field over a three-dimensional region with its volume form. In spherical polar coordinates the Euclidean volume form is $r^2\sin \theta drd\theta d\varphi$. The divergence theorem relates a volume integral of a divergence to a boundary flux integral.

A vector field preserves a chosen volume form $\omega$ when its local flow pulls $\omega$ back to itself. Infinitesimally this is ${\mathcal{L}}_X\omega =0$, using the Lie derivative. The condition depends on the chosen volume form.