grandes-ecoles 2014 QIVB

grandes-ecoles · France · centrale-maths1__pc Matrices Linear Transformation and Endomorphism Properties

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.
What is the necessary and sufficient condition on $f$ for proposition $(\mathcal{P})$ to hold?

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.

What is the necessary and sufficient condition on $f$ for proposition $(\mathcal{P})$ to hold?

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