By Frédéric Pham
Los angeles première partie de ce livre a pour yet de consolider les acquis essentiels du calcul différentiel de licence. Elle a été profondément remaniée pour cette deuxième édition. los angeles deuxième partie présente l. a. théorie intrinsèque des variétés (avec comme objectif essentiel los angeles compréhension des notions de fibré tangent et fibré general) et enchaîne sur les premiers rudiments de los angeles topologie algébrique (homotopie et revêtements). Elle se termine par une ébauche de théorie de l'intégration sur les variétés, où l'on fait connaissance avec l'homologie et los angeles cohomologie. Parallèlement aux chapitres proprement dits, qui cherchent à présenter de façon cohérente les suggestions formant l'ossature de los angeles théorie, les «études» insérées entre les chapitres ont pour yet de motnrer ces strategies en motion dans un contexte. Allant de los angeles géométrie algébrique élémentaire à los angeles mécanique, ces contextes ont été choisis de façon à donner une snapshot huge et ouverte de ce qu'est los angeles géométrie.
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Additional info for Géométrie et calcul différentiel sur les variétés : Cours, études et exercices pour la maîtrise de mathématiques
Sample text
7) generalize, for j = 1, . . , k , to 0 → εj R → pj → K[ε1 , . . , εj , . . 14) where a “hat” means “omit this coefficient”. 7) is, for any choice 1 ≤ j1 < . . < j ≤ k , 0 → i=1 εji R → R pji −→ i=1 K[ε1 , . . , εji , . . , εk ] → 0. 15) In particular, for the “maximal choice” ji = i , i = 1, 2, . . , k , we get 0 → ε1 . . εk R → R → k i=1 K[ε1 , . . , εj , . . , εk ] → 0. 16) corresponds to the “most vertical” bundle ε1 . . εk T M → T kM → k i=1 T k−1 M. There are also various projections T k K → T K for all = 0, .
Assume now that the formula holds for k ∈ N. , f (x) = f (x) ∈ W for all x ∈ U . In the following proof we will use only these properties of f . 17). Note that we used df (x)v = ∂v f (x) = ∂v f (x) = df (x)v , which holds since x, v ∈ V and f |U = f . Now we are going to repeat this argument, using that, if f is C 3 over T T K, then all maps ∂u f , u ∈ T T U , are C 2 over T T K and hence also over T K: f (x + ε1 v1 + ε2 v2 + ε1 ε2 v12 ) = f (x + ε1 v1 + ε2 (v2 + ε1 v12 )) = f (x + ε1 v1 ) + ε2 (∂v2 +ε1 v12 f )(x + ε1 v1 ) = f (x) + ε1 ∂v1 f (x) + ε2 ∂v2 +ε1 v12 f (x) + ε1 ε2 ∂v1 ∂v2 +ε1 v12 f (x) = f (x) + ε1 ∂v1 f (x) + ε2 ∂v2 f (x) + ε1 ε2 (∂v12 + ∂v1 ∂v2 )f (x).
Sections of T M are also called vector fields, and we also use the classical notation X(M ) for Γ(T M ) . In a chart (Ui , ϕi ) , vector fields can be identified with smooth maps Xi : V ⊃ ϕi (Ui ) → V , given by ∼ Xi := pr2 ◦T ϕi ◦ X ◦ ϕ−1 : ϕ−1 i i (Ui ) → Ui → T Ui = Ui × V → V. Similarly, sections of an arbitrary vector bundle are locally represented by smooth maps Xi : V ⊃ ϕ−1 i (Ui ) → W. If the chart (Ui , ϕi ) is fixed, for brevity of notation we will often suppress the index i and write the chart representation of X in the form U → U × W, x → x + X(x) or (x, X(x)).