Hi I'm trying to prove that if a vector field $F$ is the saquare of a functions $f \in C^1$ then is conservative. More precisely if $f : U \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^1$ then the vector field $F:U \to \mathbb{R}^2$ given by $F(x,y) = (f(x,y), f(x,y))$ is conservative. Naively I tried to let $\varphi(x,y) = \int_a^xf(t,y)dt+\int_b^yf(x,t)dt $ for some $(a,b) \in U$ to be the potential of $F$ but that fail for $\frac{\partial \varphi}{\partial x} = f(x,y) + \frac{\partial}{\partial x}(\int_b^yf(x,t)dt) \neq f(x,y)$
I would appreciate any help to find the potential for $F$
I've edited my question because I notice that with two distinct functions this is not true but I'm still interested to answer this question for the case f=g