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Titre: Mathématiques pour l’ingénieur

Auteurs: Daniel BOICHU, Vincent ROBIN

Ecole: Néant

Résumé: Je me rappelais la confidence de ce chef m’avouant que s’il entendait, ce qui s’appelle entendre, tous les sons de son orchestre, il deviendrait fou. Heureusement, il n’entendait que la ligne générale dans laquelle chaque instrument trouvait sa hauteur exacte. De même ne prêtais-je l’oreille à la couleur d’un timbre qu’après m’être assuré qu’il s’intégrait dans l’ensemble. Michel Del Castillo, la Tunique d’infamie. (1997) La géométrie élémentaire, mais aussi l’analyse fonctionnelle, sont d’excellents apprentissages du raisonnement. En théorie tout le monde s’accorde à dire combien il est crucial pour un citoyen de savoir raisonner. C’est la voie obligée pour participer à une société démocratique. En pratique, c’est une autre histoire, faire l’opinion est devenue une spécialité des _ communicants _. Mais ces communicants ont besoin d’une matière meuble, ce n’est pas celle des hommes formés au raisonnement et à l’autonomie de penser. Le débat et la polémique oiseuse autour du sujet de mathématique du Bac S 2003 est à méditer à l’aune de ce qui précède : mathématique formatrice ou mathématique creuse ?

Extrait du sommaire:

Préliminaires 5
Histoire à suivre 5
L’intégrale de Lebesgue, par Lebesgue en personne 6
1 L’intégrale utile du scientifique 8
1.1 Premier niveau de la théorie de l’intégration : intégrale de Riemann 8
1.1.1 Définition de l’intégrale des fonctions en escalier 8
1.1.2 Définition de l’intégrale des fonctions bornées sur des intervalles bornés 8
1.1.3 Approximation de l’intégrale par des sommes de Riemann 8
1.2 Deuxième niveau de la théorie de l’intégration : intégrale de Lebesgue 9
1.2.1 Vers la définition 9
1.2.2 L’exemple lumineux : la fonction de Dirichlet 11
1.2.3 L’intégrale de Lebesgue sur R 11
1.2.4 Intégrale sur une partie mesurable de R 12
1.2.5 L’intégrale sur Rn 12
1.2.6 L’intégrale de Lebesgue sur un espace mesuré 13
1.2.7 Exemples (très instructifs) 13
1.2.8 Ensemble négligeable 14
1.3 Consistance entre les intégrales 14
1.4 Troisième niveau de la théorie de l’intégration : l’intégrale impropre 15
1.4.1 Exemple de base 15
1.4.2 Remarque de terminologie 15
1.4.3 Plus généralement 15
1.4.4 Deux exemples incontournables 15
1.4.5 Remarque 16
1.5 Espaces fonctionnels classiques 16
1.5.1 Notations 16
1.5.2 Mesure de comptage 17
1.6 Exemples (variés) 17
1.6.1 Exemple 1 17
1.6.2 Exemple 2 17
1.6.3 Exemple 3 18
1.6.4 Exemple 4 18
1.6.5 Exemple 5 18
1.7 Théorèmes d’interversion 18
1.7.1 Théorème de convergence monotone (Levi 1906) 19
1.7.2 Théorème de convergence dominée (Lebesgue) 19
1.8 Intégrale dépendant d’un paramètre 20
1.8.1 Théorème (continuité de F) 20
1.8.2 Théorème (de dérivation sous le signe somme) 21
1.8.3 Importance de ces théorèmes 22
2 La base des distributions 23
2.1 Introduction heuristique 23
2.2 Fonction comme fonctionnelle 23
2.2.1 Introduction 23
2.2.2 D(R) ou la classe des fonctions test sur R 23
2.3 Définition d’une distribution 24
2.3.1 Définition 24
2.4 Dérivée d’une distribution 25
2.5 Multiplication fT quand f est C1 26
2.6 Support d’une distribution 26
2.7 Convergence d’une suite Tn de distributions 26
2.8 Primitive d’une distribution sur R 27
2.9 Exemples de distributions 27
2.9.1 Distribution sur R2 (ou R3) portée par une courbe orientée 27
2.9.2 Distribution portée par une surface 28
3 Fourier périodique 29
3.1 Heuristique 29
3.2 La clef 30
3.3 Rappel jusqu’à Banach et Hilbert 30
3.3.1 Espace métrique (X, d) 30
3.3.2 Espace vectoriel normé (E, k k) 30
3.3.3 Espace de Banach 31
3.3.4 Produit scalaire sur le R ou C e.v. E 31
3.4 Représentation de Fourier dans L2([0, T[) 32
3.4.1 On revient sur quelques définitions 32
3.4.2 Fondamental 33
3.5 Fourier dans L1([0, T]) 33
3.6 Une condition su sante de convergence ponctuelle de la série de Fourier 34
3.7 Le miracle des distributions périodiques 36
4 La convolution 39
4.1 Convolution dans les fonctions définies sur R 39
4.1.1 Convolution, dérivation et translation 40
4.2 Convolution dans les fonctions périodiques 40
4.3 Convolution dans les suites 41
4.4 Convolution entre une distribution et une fonction 42
4.5 Convolution entre distributions 42
4.5.1 Produit tensoriel entre distributions 42
4.5.2 Définition de T S 43
4.5.3 Propriétés 43
4.6 L’algèbre de convolution D0 44
4.6.1 Théorème du support 44
4.6.2 Lois de composition interne 44
4.7 Régularisation d’une distribution 45
5 Transformées de Fourier et de Laplace 46
5.1 Analyse de Fourier 46
5.1.1 Transformée de Fourier 46
5.1.2 Autres définitions 47
5.2 Synthèse de Fourier 47
5.2.1 L’information spectrale 47
5.2.2 Convergence en valeur principale 48
5.3 Transformée de Fourier dans Rn 48
5.4 Les charmes de la transformée de Fourier 48
5.4.1 Echange translation – modulation 48
5.4.2 Echange dérivation – multiplication monômiale 50
5.4.3 Echange multiplication – convolution 50
5.4.4 Changement d’échelle 50
5.5 Fourier dans L2(R) 51
5.6 Espace S0(R) des distributions tempérées 51
5.6.1 Espace de Schwartz S 51
5.6.2 Comment étendre la définition de la transformée de Fourier ? 52
5.6.3 Distribution tempérée 52
5.7 F(T) ou ˆ T quand T 2 S0 53
5.8 Des exemples 53
5.9 Transformée de Laplace 54
5.9.1 Exemples 55
5.9.2 Comportement de la transformée de Laplace 55
5.9.3 Exemple travaillé 56
Epilogue 57
L’intégrale de Lebesgue en mathématique 57
La fin de l’histoire ? 57
Références bibliographiques 58

Mathématique appliquée cours 20

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