grandes-ecoles 2014 QIIIB1

grandes-ecoles · France · centrale-maths1__pc Implicit equations and differentiation Gradient computation for multivariable implicit/explicit functions

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 for all $x$ in $\mathbb{R}^n$, and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.

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 for all $x$ in $\mathbb{R}^n$, and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.

Paper Questions

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