grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2014 centrale-maths1__pc

26 maths questions

QIA Matrices Linear Transformation and Endomorphism Properties View
Let $A$ be a real square matrix of size $n$ and $b$ an element of $\mathbb{R}^n$. Let $f$ be the map from $\mathbb{R}^n$ to $\mathbb{R}^n$ defined by $$\forall x \in \mathbb{R}^n \quad f(x) = Ax + b$$ Show that $f$ is of class $C^1$ and specify its Jacobian matrix $J_f(x)$ at every point $x$ of $\mathbb{R}^n$.
QIB1 Chain Rule Chain Rule with Composition of Explicit Functions View
Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.
QIB2 Taylor series Taylor's formula with integral remainder or asymptotic expansion View
Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Deduce that in a neighbourhood of 0 $$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$
QIC1 Matrices Determinant and Rank Computation View
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 $j$.
We admit that if functions $\varphi_1, \varphi_2, \ldots, \varphi_n$ are continuous on $\mathbb{R}$ and take values in $\mathbb{R}^n$, then the function $\Phi$ defined on $\mathbb{R}$ by: $$\Phi(t) = \operatorname{det}(\varphi_1(t), \varphi_2(t), \ldots, \varphi_n(t))$$ is continuous on $\mathbb{R}$.
Using question I.B.2 and the multilinearity of the determinant, show that in a neighbourhood of 0 $$\operatorname{det}\left(f(t_1), f(t_2), \ldots, f(t_n)\right) = t^n \mathrm{jac}_f(0) + \mathrm{o}\left(t^n\right)$$
QIC2 Matrices Determinant and Rank Computation View
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 $$\lim_{t \to 0} \frac{\operatorname{det}\left(f(t_1), \ldots, f(t_n)\right)}{\operatorname{det}\left(t_1, \ldots, t_n\right)} = \mathrm{jac}_f(0)$$
QIC3 Matrices Determinant and Rank Computation View
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).
QIIA Matrices Linear Transformation and Endomorphism Properties View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$.
For $x$ in $\mathbb{R}^2$, express $\operatorname{div}_f(x)$ using only $A$.
QIIB1 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
What is $u_a(t)$?
QIIB2 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Let $a$ and $b$ be two elements of $\mathbb{R}^2$ and let $t$ be a real number. Show that $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$
QIIB3 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Use the result of II.B.2 to interpret the sign of $\operatorname{div}_f(a)$ in terms of the direction of variation of the area of a certain parallelogram as a function of $t$.
QIIC1 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
We set $a = (a_1, a_2)$ and $u_a(t) = (x_1(t), x_2(t))$. We assume that $\lambda_1 \neq 0$ and $a_1 > 0$. Determine a function $\theta_a$ such that $x_2(t) = \theta_a(x_1(t))$ for every real $t$.
QIIC2 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
In this question, $a = (2,1)$ and $b = (1,2)$.
For each of the following cases, illustrate on the same figure the graphs of the functions $\theta_a$, $\theta_b$ and $\theta_{a+b}$, as well as the parallelograms with vertices $(0,0)$, $u_a(t)$, $u_b(t)$ and $u_a(t) + u_b(t)$ for $t = 0$ and a strictly positive value of $t$.
a) $\lambda_1 = 1$ and $\lambda_2 = 2$.
b) $\lambda_1 = 1$ and $\lambda_2 = -2$.
c) $\lambda_1 = 1$ and $\lambda_2 = -1$.
QIID1 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Revisit questions II.B.1 and II.B.2 in the case where $A$ is triangular of the form $$A = \begin{pmatrix} \lambda & \mu \\ 0 & \lambda \end{pmatrix}$$
QIID2 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Show that the relation $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ holds when the matrix $A$ has a characteristic polynomial that splits over $\mathbb{R}$.
QIID3 Second order differential equations Reduction of a differential system to a second-order ODE View
We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Extend the result $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ to the case of an arbitrary real $2 \times 2$ matrix.
QIIIA Implicit equations and differentiation Gradient computation for multivariable implicit/explicit functions View
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 $f_{i,j,k}(x) = f_{j,i,k}(x)$.
QIIIB1 Implicit equations and differentiation Gradient computation for multivariable implicit/explicit functions View
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 $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.
QIIIB2 Implicit equations and differentiation Gradient computation for multivariable implicit/explicit functions View
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 $\mathbb{R}^n$.
Deduce that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = 0$.
QIIIB3 Implicit equations and differentiation Differentiability proof and derivative formula for abstract/matrix-valued functions View
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 there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.
QIIIB4 Implicit equations and differentiation Differentiability proof and derivative formula for abstract/matrix-valued functions View
Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. What is the necessary and sufficient condition on $f$ for the Jacobian matrix $J_f(x)$ to be antisymmetric for all $x$ in $\mathbb{R}^n$?
QIIIC Implicit equations and differentiation Differentiability proof and derivative formula for abstract/matrix-valued functions View
Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself.
Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that $$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$
One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.
QIVA1 Matrices Matrix Algebra and Product Properties View
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.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = \alpha_{i,k,j} = -\alpha_{k,j,i}$.
QIVA2 Matrices Matrix Algebra and Product Properties View
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.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Deduce that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = 0$.
QIVA3 Matrices Linear Transformation and Endomorphism Properties View
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.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that there exist an orthogonal matrix $A$ and an element $b$ of $\mathbb{R}^n$ such that, for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$.
One may interpret the relations $\alpha_{i,j,k} = 0$ using matrix products.
QIVB Matrices Linear Transformation and Endomorphism Properties View
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?
QIVC Matrices Linear Transformation and Endomorphism Properties View
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.
If $g$ is a function of class $C^2$ from $\mathbb{R}^n$ to $\mathbb{R}$, we denote $\Delta_g(x) = \sum_{i=1}^n \frac{\partial^2 g}{\partial x_i^2}(x)$ (Laplacian of $g$ at $x$). Show that $(\mathcal{P})$ is equivalent to the proposition $$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$