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Let $\dot x= f(x,y)$ and $\dot y=g(x,y)$ be continuously differentiable vectorfield on $\mathbb R^2$ with flow $\varphi_t$. Show that $\int\int\limits_A\left(\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}\right) dx dy=0$ for the integral of the divergence of the vectorfield over area $A$ from question 1.

So in question 1 they defined open area $A\subseteq \mathbb R^2$ from which the border is a periodic orbit. In question 1 I showed that $\varphi_t(A)=A$ and $\varphi_t(\bar A)=\bar A$, where $\bar A$ is the border of $A$.

I don't know where to start with this question. Any hints are much appreciated !

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You could be inspired by the divergence theorem, the integral that you have corresponds to the integral of $\nabla\cdot(f,g)$ and your results form question 1 indicate that $(f,g)\cdot \mathbf{n}$ will vanish.

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