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Titre: Méthodes Mathématiques pour la Licence de Physique et Chimie

Auteurs: Jean-Luc Raimbault

Ecole: Laboratoire de Physique des Plasmas, Ecole Polytechnique

Résumé: Les pages qui suivent présentent quelques méthodes mathématiques que vous aurez à utiliser dans vos cours de Physique et Chimie. Cet enseignement de Mathématique est structuré en 5 grandes parties :

– Variables complexes
– Equations différentielles
– Analyse dans Rn
– Algèbre linéaire
– Analyse de Fourier.

Au sein de chacune de ses parties, plusieurs chapitres, allant du plus simple au plus compliqué, sont proposés. Les chapitres 1, 4, 5, 6, 9, 10, 12, 14 et 15 (cf. sommaire) seront traités en première intention et devraient être maitrisés par tous les étudiants. Ils constituent la base du programme sur lequel vous serez interrogés. Les chapitres complémentaires, qui abordent des notions plus avancées, peut-être moins utiles pour certains d’entre vous, seront étudiés si le temps le permet et/ou proposés aux étudiants suffisamment à l’aise sur les chapitres de base.

Mettre en oeuvre des méthodes mathématiques dans le contexte d’un problème de Physique ou Chimie suppose une connaissance des concepts mathématiques associés (ce que ce cours vous rappellera ou vous fera découvrir), et surtout une mise en pratique qui passe par la résolution de nombreux exercices. Cela suppose une présence assidue et active aux cours et aux travaux dirigés mais également un travail personnel important. Cette implication personnelle est déterminante et sera encouragée. L’objectif est d’arriver progressivement à identifier vos lacunes, puis à travailler – avec notre aide – à les combler, enfin à estimer par vous-même le niveau de compréhension que vous avez atteint. Pour vous y aider, des devoirs et tests vous seront régulièrement proposés et des livres d’exercices seront à votre disposition.

Enfin, il est bon de rappeler qu’un bagage mathématique s’entretient. Il vous faut donc prévoir de revenir périodiquement, tout au long de vos études (et même après !) sur des concepts et des méthodes que vous maitriserez d’autant moins que vous les utiliserez de façon occasionnelle. Les livres sont faits pour ¸ca. Les quelques indications suivantes pourront éventuellement vous guider dans la jungle des ouvrages disponibles.

Extrait du sommaire:

I Variables complexes 5
1 Rappels sur les nombres complexes 7
1.1 Motivations historiques 7
1.2 Le corps des nombres complexes 8
1.3 Exponentielle complexe 9
1.4 Représentation des nombres complexes 10
1.4.1 Représentation cartésienne 10
1.4.2 Représentation géométrique 10
2 Fonctions d’une variable complexe 13
2.1 Fonctions d’une variable complexe 13
2.2 Dérivation des fonctions d’une variable complexe 14
2.3 Intégration des fonctions d’une variable complexe 15
2.4 Développement en série 17
2.5 Méthode des résidus 18
3 Transformation de Laplace au sens des fonctions 21
3.1 Introduction 21
3.2 Définition de la transformée de Laplace 22
3.3 Holomorphie 23
3.4 Propriétés de la transformée de Laplace 24
3.5 Comportements asymptotiques 26
3.6 Inversion de la transformée de Laplace 27
3.7 Remarque 30
3.8 Exercices 30
II Equations différentielles 35
4 Introduction 37
4.1 Mécanique. 37
4.2 Dynamique des Populations. 38
4.3 Equations aux Dérivées Partielles. 39
5 Rappels sur les équations différentielles 41
5.1 Terminologie 41
5.2 Quelques conséquences de la linéarité 42
5.3 Deux solutions explicites importantes 42
5.3.1 Equation différentielle linéaire du 1er ordre 43
5.3.2 Equations différentielles du second ordre à coefficients constants 44
5.4 Système d’équations différentielles du 1er ordre 46
5.5 Equation intégrale 48
5.6 Théorème d’existence et d’unicité 49
6 Systèmes Différentiels Linéaires 51
6.1 Exponentielle de matrice 51
6.2 Propagateur 52
6.3 Calcul pratique du propagateur 53
6.4 Equations différentielles en dimension 2 54
6.4.1 Diagonalisation des matrices 2 × 2 54
6.4.2 Forme explicite du propagateur en dimension 2 55
7 Analyse qualitative des équations différentielles 59
7.1 Exemple 59
7.2 Classification des points fixes (1 dimension) 60
8 Stabilité des systèmes différentiels 63
8.1 Stabilité des systèmes différentiels linéaires 63
8.1.1 Portraits de phase en dimension 2 63
8.2 Stabilité des systèmes différentiels non linéaires 65
8.2.1 Linéarisation 65
8.2.2 Cycles limites 67
III Analyse dans Rn 69
9 Différentier et Intégrer 71
9.1 Dérivées et différentielles 71
9.2 Circulation et flux 73
9.3 Opérateurs différentiels 76
9.4 Formule de Stokes 78
9.5 Singularités 79
9.6 Différentielles exactes 81
9.7 Théorèmes de Helmholtz 82
9.8 Théorème de Leibnitz 83
10 Approximer et calculer 85
10.1 Formule de Taylor 85
10.2 Approximation de Stirling 86
10.3 Distribution de Dirac 87
10.4 Intégrales Gaussiennes 89
11 Minimiser 91
11.1 Extrémalisation sans contraintes 91
11.2 Multiplicateurs de Lagrange 92
11.3 Transformation de Legendre 94
IV Algèbre linéaire 97
12 Eléments d’algèbre linéaire 99
12.1 Déterminants 99
12.2 Matrices 100
12.2.1 Opérations 100
12.2.2 Inversion 101
12.3 Système d’équations linéaires 102
12.4 Diagonalisation 103
13 Scalaires, vecteurs, tenseurs 107
13.1 Introduction 107
13.2 Notations et changement de base 108
13.3 Scalaires, vecteurs et tenseurs 109
13.4 Produits et contractions 110
13.5 Base réciproque 112
V Analyse de Fourier 115
14 Séries de Fourier 117
14.1 Des problèmes historiques 117
14.2 Définition des séries de Fourier 118
14.3 Convergence des séries de Fourier 119
14.4 Dérivation et Intégration terme à terme 119
14.5 Phénomène de Gibbs 120
14.6 Exercices 121
15 Transformation de Fourier au sens des fonctions 123
15.1 Motivations 123
15.2 Transformée de Fourier dans L1(R) 125
15.3 Dérivation et Inversion 127
15.4 Convolution 129
15.5 Transformée de Fourier dans L2(R) 130
15.6 Transformées de Fourier dans S(R) 131
15.7 Transformées à plusieurs variables. 132
15.8 Applications aux EDO à coefficients constants 133
15.9 Conclusion 136

Mathématique appliquée cours 4

Télécharger le fichier PDF: Méthodes Mathématiques pour la Licence de Physique et Chimie

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