grandes-ecoles 2014 QIC2

grandes-ecoles · France · centrale-maths1__pc Matrices Determinant and Rank Computation

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
Deduce that $$\lim_{t \to 0} \frac{\operatorname{det}\left(f(t_1), \ldots, f(t_n)\right)}{\operatorname{det}\left(t_1, \ldots, t_n\right)} = \mathrm{jac}_f(0)$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.

Deduce that
$$\lim_{t \to 0} \frac{\operatorname{det}\left(f(t_1), \ldots, f(t_n)\right)}{\operatorname{det}\left(t_1, \ldots, t_n\right)} = \mathrm{jac}_f(0)$$

Paper Questions

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