I am trying to go about a proof of Poincaré's Lemma in analysis, and I am having some difficulty with the proof. I would like to prove that if $U \subset \mathbb{R}^n$ be an open set that is starshaped with respect to a point $\epsilon$ and let $\mathbf{g} : U \rightarrow \mathbb{R}^n$ be an irrotational vector field of class $C^1$, Then $\mathbf{g}$ is a conservative vector field.
I initialized this proof by setting $$f(\mathbf{x}) = \int_{\gamma} \mathbf{g}, \mathbf{x} \in U,$$ where $\gamma$ is the cure given by the parametric representation $\phi : [0,1] \rightarrow \mathbf{R}^n$ defined by $\phi(t) = \epsilon + t(\mathbf{x} + \epsilon)$. My friend then recommended that I recognize that $$f(\mathbf{x}) = \int_0^1 \sum_{j=1}^n g_j(\epsilon + t(\mathbf{x} + \epsilon))(x_j - x_{0,j}) dt,$$ I am having difficulty seeing where I can get this result. Any recommendations on how to prove the previous statement, and how I may continue to finish this proof?