(i) Let $\omega$ be a closed $1$-form in $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$ satisfying: $$\int_{\gamma}\omega=0,$$ where $\gamma(t) = (\cos t,\sin t)$, $t \in [0, 2\pi]$. Show that $\omega$ is exact in $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$.}
(ii) Show that any closed $1$-form $\omega\in \mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$ decomposes as: $$\omega = \lambda\omega_{0} + \alpha,$$ where $\lambda\in\mathbb{R}$, $\alpha$ is exact in $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$, and $\omega_{0}$ is the (closed) ‘angle $1$-form’ in $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$: $$\omega_{0}=-\frac{y}{x^{2}+y^{2}}dx+\frac{x}{x^{2}+y^{2}}dy.$$
I am trying to work through this problem, but I think my (lack of) understanding of differential forms is holding me back.
So for the first question, I want to show that $\int_{\gamma}\omega=0$ for any closed curve $\gamma$ $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$. Clearly this is true for any path that is homotopic to $S^{1}$. I feel that I should try to show that $\int_{\gamma}\omega=0$ has some uniform bound for any $\gamma$, but I am not sure how to construct it.
Also, I am confused how to actually look at this line integral and manipulate it.
The definition of line integral says that $$\int_{\gamma}\omega=\int_{0}^{2\pi}(\omega(\gamma(t))[\gamma'(t)]dt$$ and I have seen another definition that says $$\int_{\gamma}\omega=\int_{0}^{2\pi}(\omega(\gamma(t)), \gamma'(t))dt$$ Here I am mostly confused how to manipulate $\omega$. In coordinates $\omega=\sum_{i}^{n}a_{i}(x)dx_{i}$ and $\omega$ is a map from $U\subset\mathbb{R}^{2}\rightarrow \Lambda^{1}$, where $\Lambda^{1}$ is the space of linear functionals $\mathbb{R}^{n}\rightarrow \mathbb{R}$. So should I be looking at $\omega$ as a scalar? I do not understand how the decomposition of $\omega$ is used. My notes say nothing about each $a_{i}$ other than the fact that $\omega$ is $C^{k}$ if each $a_{i}$ is $C^{k}$
Given the two definitions of line-integral, I have either
$$\int_{\gamma}\omega=\int_{0}^{2\pi}\omega(\gamma(t)) [\gamma'(t)]dt = \int_{0}^{2\pi}\omega(\cos t, \sin t)[(-\sin t, \cos t)]dt=0$$
$$\int_{\gamma}\omega=\int_{0}^{2\pi}(\omega(\gamma(t)), \gamma'(t))dt = \int_{0}^{2\pi}(\omega(\cos t, \sin t), (-\sin t, \cos t))dt=0$$ But to me, neither of these integrands make sense.
The theorems we have to show exactness relate to being starshaped or simply connected, but clearly $\mathbb{R}^{2}\setminus\left\{\mathbf{0}\right\}$ is neither because of the hole at the origin.
We also know that since $\omega$ is closed, that $$\frac{\partial a_{i}}{x_{j}}-\frac{\partial a_{j}}{x_{i}}=0,\ \ \forall i\neq j$$ but I am not sure how to relate this to the line integral.
I am unsure how to approach part (ii).
We have covered a lot of Lebesgue integration, $L^{p}$ spaces, convergence theorems, etc. in my intro to analysis course, all of which I mostly understood well, but differential forms have been over my head. I appreciate any hints to help me understand the problems better.