grandes-ecoles 2014 QIIIA

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 $\mathrm{D}_{i,j} f_k(x)$ or $\frac{\partial^2 f_k}{\partial x_i \partial x_j}(x)$, or also $f_{i,j,k}(x)$.
Justify that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = f_{j,i,k}(x)$.

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 $\mathrm{D}_{i,j} f_k(x)$ or $\frac{\partial^2 f_k}{\partial x_i \partial x_j}(x)$, or also $f_{i,j,k}(x)$.

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

Paper Questions

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