I am reading "Methods of information geometry by shun-Ichi-Amari" and I got stuck in the following chapter 3 section 3.3(Dually flat spaces), let me define it first
Let $(S,g,\nabla,\nabla^*)$ is dually flat space then there exits the $\nabla-$affine coordinate system $[\theta^i]$ and $\nabla^*-$affine coordinate system $[\eta_{j}]$ such that $\langle \; \partial_i, \partial^{j}\; \rangle=\delta_{i}^{j}$.
Let the component of the metric $g$ with respect to $[\theta^i]$ and $[\eta_{j}]$ we have $$g_{ij}=\langle \;\partial_{i} ,\partial_{j} \; \rangle \text{ and } \langle \;\partial^{i} ,\partial^{j} \; \rangle$$
Consider the following partial differential equation for a function $\psi:S\to\mathbb{R}$: $$\partial_{i}\psi=\eta_{i}.........................(1)$$ we may rewrite this as $d\psi=\eta_{i}d\theta^{i}$ and a solution existes if and only if $\partial_{i}\eta_{j}=\partial_{j}\eta_{i}$
My question is how we can write $(1)$ as $d\psi=\eta_{i}d\theta^i$? and why solution exists when $\partial_{i}\eta_{i}=\partial_{j}\eta_{j}?$ Can someone explain it please? Thanks.