Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself. Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that $$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$ One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.
Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself.
Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that
$$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$
One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.