Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} \rightarrow \mathbb{R}^n$; $(s,t) \mapsto b(s,t)$ by
$$b(s,t)=\Phi_s(c(t)) $$
note $s \in [0,1], t=(t^1,\dots,t^k) \in [0,1]^k$. Let $\omega$ be a $k$-form on $\mathbb{R}^n$ such that
$$L_{\mathbb{Y}}\omega=0, i_{\mathbb{Y}}\omega=0 $$
Show that
$$\int_b d\omega=0 $$
I think it would wise to use Stoke's theorem and somehow the identity $\displaystyle \frac{\partial}{\partial t}\hat{\Phi}_t \omega=\hat{\Phi}_t^*L_{\hat{\mathbb{X}}_t}\omega$.
I started
$$\begin{align} \int_b d\omega &= \int_{\partial b} \omega \\ &= \sum_{j=1}^k \sum_{\alpha=0,1} \int_{b_{(j,\alpha)}} \omega \\ &= \sum_{j=1}^k \sum_{\alpha=0,1} \int_{I^k} b^*_{(j,\alpha)} \omega \\ &= ?? \end{align}$$
Could someone please show me how to proceed with this question.
Thanks