Let $\eta = \sum_{i=1}^n a_i(\mathbf x)dx_i$ be a closed, $C^2$ $1$-form defined in a convex open set $E$ of $\mathbb{R}^n$. Prove that it is exact by following the outline:
For $\mathbf p \in E$, define $$f(\mathbf x) := \int_{[\mathbf p,\mathbf x]}\eta \qquad (\mathbf x \in E)$$ and apply Stokes' theorem to the oriented affine $2$-simplex $[\mathbf p, \mathbf x, \mathbf y]^1$.
Deduce that $$f(\mathbf y) - f(\mathbf x) = \sum_{i=1}^n({y_i-x_i})\int_{0}^{1}{a_i((1-t)\mathbf x + t\mathbf y)dt}$$ for $\mathbf x, \mathbf y \in E$. Hence $(D_if)(\mathbf x) = a_i(\mathbf x)$.
Assume that $\eta$ is a $1$-form such that $$\int_\gamma \eta = 0$$ for every $C^1$ closed curve $\gamma$. Prove that $\eta$ is exact by imitating steps in the last point.
Assume $\eta$ is a closed 1-form in $\mathbb{R}^3\setminus \{(0,0,0)\}$. Prove that it is exact.
The last question looks like I should use Poincare's lemma, except that it is punctured at the origin. I want to try the first question, but I cannot understand the notation being used here, even after carefully reading Baby Rudin's chapter 10 (Background: I have taken Analysis with Rudin's book, chapters 1~10). Any help would be much appreciated.
$^1$ An oriented affine $k$-simplex is defined in Baby Rudin ch. 10, page 266, as follows: $\sigma = [\mathbf p_0, \mathbf p_1, \dots, \mathbf p_k]$ is the affine mapping $\sigma : \mathbb R^k \to \mathbb R^k$ such that $\sigma( \mathbf 0) = \mathbf p_0$ and $\sigma( \mathbf e_i) = \mathbf p_i$ for $i=1,\dots,k$.