MTH-696A: Topics in Geometric Mechanics Assignment 1 (by Dr. Leo Butler)
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-
Let $\crossproduct : \R^3 \times \R^3 \to \R^3$ be the vector (=cross)
product. Define the following operator $\omega$
\begin{equation}\label{eq:omega}
\omega_x ( u, v ) := \ip{x}{u \crossproduct v},
\end{equation}
for $x \in \R^3$ and $u, v \in T_x \R^3 \equiv \R^3$. Let $e_1,
e_2, e_3$ be the standard basis of $\R^3$ and let $x_i$ be
coordinates induced by this basis (so $x=x_1 e_1 + x_2 e_2 + x_3
e_3$).
-
Compute $\omega = \sum_{i \lt j} \omega_{ij}(x) \D{x_i} \wedge \D{x_j}$.
-
Compute the exterior derivative of $\omega$, $\D{\omega}$, and show
that this equals $\D{x_1} \wedge \D{x_2} \wedge \D{x_3}$, the
standard volume element on $\R^3$.
Let $\sphere{2} = \set{x \in \R^3 \st \norm{x}=1}$ be the unit
sphere. Let $\eta = \omega |_{\sphere{2}}$ be the restriction of the
$2$-form $\omega$ defined in (\ref{eq:omega}). Show that $\eta$ is a
closed, non-degenerate $2$-form.
