Mathematics Asked by Lorenzo Andreaus on December 15, 2020

Given $dcolon [0,1]to mathbb{R}^2$, with $d(0)=d(1)=0$, I want to find a function $ccolon [0,1]^2to mathbb{R}^2$ such that $c(x,0)-c(1,0)+c(0,1)-c(x,1)+c(1,x)-c(0,x)=d(x)$, for all $xin [0,1]$.

I was thinking about consider $c(x,0)-c(1,0)$, $c(0,1)-c(x,1)$ and $c(1,x)-c(0,x)$ as the integral of $c$ along some paths in $mathbb{R}^2$, and think of $c$ as some kind of a potential function, but I don’t know how to proceed.

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