I read the section closed forms and cycles in Arnold's Mathematical Methods of Classical Mechanics (page 196-200), but the problems in this section is too difficult to solve in the way following the hints. What's more, There seems no other materials teach closed forms and cycles in the same way as this book. So, this post is meant to ask for solutions(in the way following the hints) and recommended materials. The problems are following:
Let $X$ be a vector field on $M$ and $\omega$ a differential $k$-form. We define a $(k-1)$-form $i_X\omega$ (the interior derivative of $\omega$ by $X$) by the relation $$ (i_X \omega)(\xi_1,\ldots,\xi_{k-1}) = \omega(X,\xi_1,\ldots,\xi_{k-1}). $$ Prove the homotopy formula $$ i_X d + di_X=L_X. $$
Prove the formula for differentiating a vector product on three dimensional Euclidean space (or on a Riemannian manifold) : $$ \mathrm{curl}[a,b] = \{a,b\} + a~\mathrm{div}[b] - b~\mathrm{div}[a], $$ where $\{a,b\} = L_ab$ is the Poisson bracket of the vector field.