Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal. If $g$ is a function of class $C^2$ from $\mathbb{R}^n$ to $\mathbb{R}$, we denote $\Delta_g(x) = \sum_{i=1}^n \frac{\partial^2 g}{\partial x_i^2}(x)$ (Laplacian of $g$ at $x$). Show that $(\mathcal{P})$ is equivalent to the proposition $$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$
Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
If $g$ is a function of class $C^2$ from $\mathbb{R}^n$ to $\mathbb{R}$, we denote $\Delta_g(x) = \sum_{i=1}^n \frac{\partial^2 g}{\partial x_i^2}(x)$ (Laplacian of $g$ at $x$). Show that $(\mathcal{P})$ is equivalent to the proposition
$$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$