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Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that

$\hat{\Phi}^{*}_1 \beta = \beta$, $\displaystyle \lim_{t \rightarrow 0}\hat{\Phi}_t^* \beta=0$

Then $\beta=d \alpha$ where $\displaystyle \alpha \int^1_0 \hat{\Phi}^*_t\Big(i_\hat{\mathbb{X_t}}\beta \Big)dt$

where $\hat{\mathbb{X_t}}$ is defined by $\displaystyle \frac{\partial}{\partial t}\hat{\Phi_t}(x)=\hat{\mathbb{X_t}}(\hat{\Phi_t}(x))$.

The proof in my notes starts by declaring that $ \beta= \hat{\Phi_1}^*(x)$ and $\displaystyle \lim_{t \rightarrow 0}\hat{\Phi_t}^* = \int^1_0 \frac{\partial}{\partial t}\hat{\Phi_t}^* \beta dt$

I cannot see how either of these equations hold.

Then it says using the proposition that $\frac{\partial}{\partial t}\hat{\Phi_t}^*=\hat{\Phi_t}^*L_{\hat{\mathbb{X_t}}}\omega$, that

$\frac{\partial}{\partial t}\hat{\Phi_t}^* \beta=\hat{\Phi_t}^*L_{\hat{\mathbb{X_t}}}\beta$

I cannot see where the $\beta$ comes from on the LHS.

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The first equation $\beta=\hat{Φ_1}^∗(x)$ is probably just a typo. I think you mean $\beta=\hat{Φ_1}^∗(\beta)$, which is the hypothesis. The second equation is also a typo. What you want is just use the fundamental theorem of calculus: $f(1) - f(\epsilon) = \int_{\epsilon}^1 \frac{\partial f}{\partial t} dt$ (apply it to $f(t)=\hat{Φ_t}^∗(\beta)$ and then take the limit $\epsilon\to 0$).

Finally the conclusion makes use of the Cartan formula which says, for a closed fom $\alpha$, $L_X\alpha = d(i_X\alpha)$. I'm sure you can figure out this now. You use also $dΦ^*\beta=Φ^* d\beta$, and at the last step you pull the $d$ out of the integral.

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