Let $X = (x^1, x^2, \dots, x^n)$ and $Y = (y^1,y^2,\dots,y^n)$ be coordinate systems onto a smooth manifold $M$. Denote by $\partial_i$ the i-th coordinate field of $Y$. Now, I'm trying to solve the PDE system given by
$$y^i = \partial_i f(X).$$
It is part of a proof present in the book "Information Geometry", more preciselly in the theorem 4.3, which states that
There exist strictly convex potential functions f(X) and g(Y) satisfying $$y^i = \partial_i f(X), \quad x^j = \partial_j g(Y).$$
The proof starts with the argument "the equation $y^i = \partial_i f(X)$ can be solved locally iff $$\partial_i y^j = \partial_j y^i~'',$$ but I don't understand from where this existence condition comes. Applying the PDE in this condition, I recover the Schwarz's symmetry to second derivatives. Could someone help me understand this argument?