grandes-ecoles 2014 QIC3

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 $j$.
In the case $n = 2$ (respectively $n = 3$), give a geometric interpretation of the absolute value of the Jacobian of $f$ at 0 using areas of parallelograms (respectively volumes of parallelepipeds).

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 $j$.

In the case $n = 2$ (respectively $n = 3$), give a geometric interpretation of the absolute value of the Jacobian of $f$ at 0 using areas of parallelograms (respectively volumes of parallelepipeds).

Paper Questions

QIA QIB1 QIB2 QIC1 QIC2 QIC3 QIIA QIIB1 QIIB2 QIIB3 QIIC1 QIIC2 QIID1 QIID2 QIID3 QIIIA QIIIB1 QIIIB2 QIIIB3 QIIIB4 QIIIC QIVA1 QIVA2 QIVA3 QIVB QIVC