From 345af62248ce768263a1197f54a5dddb6996dcff Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?F=C3=A9lix=20MARQUET?= Date: Mon, 2 Dec 2024 10:07:10 +0100 Subject: [PATCH] Obisidian vault auto-backup: 02-12-2024 10:07:10 on constellation. 2 files edited --- ISEN/Maths/CIPA3/Fiche de révision DS1.md | 9 ++++++--- ISEN/Maths/CIPA3/Fiche de révision DS1.pdf | Bin 153615 -> 154760 bytes 2 files changed, 6 insertions(+), 3 deletions(-) diff --git a/ISEN/Maths/CIPA3/Fiche de révision DS1.md b/ISEN/Maths/CIPA3/Fiche de révision DS1.md index d0e06d2..6424552 100644 --- a/ISEN/Maths/CIPA3/Fiche de révision DS1.md +++ b/ISEN/Maths/CIPA3/Fiche de révision DS1.md @@ -1,6 +1,5 @@ ```table-of-contents title: Fiche de révision DS1 de maths -includeLinks: false ```
@@ -114,9 +113,11 @@ $$\int_{-\infty}^\infty \delta(x - a) f(x) \, dx = f(a)$$ 4. **Lien avec la transformée de Fourier** : $$\mathcal{F}(\delta(x)) = 1$$ # 8. Distribution de 2 variables +## Définition +Une **distribution de deux variables** est une généralisation des fonctions classiques permettant de modéliser des phénomènes singuliers ou localisés, comme les impulsions ou les discontinuités. Elle agit sur des fonctions tests $\phi(x, y)$ lisses et à support compact par une intégrale généralisée. + ## Gradient d'une fonction à 2 variables $$\nabla f(x, y) = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \end{bmatrix}$$ -
## Dérivée partielle selon x $$ @@ -135,13 +136,15 @@ La matrice hessienne d'une fonction $f: R^n \to R$ est une matrice carrée compo $$ H_f(x) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} $$ +
+ ## Propriétés 1. La matrice hessienne est **symétrique** si $f$ est de classe $C^2$ $\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.$ 2. La hessienne permet de déterminer la **convexité** ou la **concavité** de $f$ : - Si $H_{f}(x)$ (x) est définie positive ($∀v, v^TH_{f}(x)v>0$) alors $f$ est **strictement convexe**. - Si $H_{f}(x)$ (x) est définie négative ($∀v, v^TH_{f}(x)v<0$) alors $f$ est **strictement concave**. -## Exemple: $f(x, y) = x^2 + xy + y^2$ +## Exemple: f(x, y) = x^2 + xy + y^2 ## Calcul des dérivées partielles 1. 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