Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$. We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$. Show that there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.
Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.