Ivan Shishkin, Rye (1878)

Problems/Differential geometryExerciseUnreviewed

Ma première forme différentielle

by Ancient Tree·
45
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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On considère la forme différentielle sur R2\R^{2} :
ω=xdx+ydy.\omega = xdx+ydy.

  1. Exprimer ω(x,y)(u,v)\omega_{(x,y)}(u,v) pour tout (u,v)(u,v) de R2\R^{2}.
  2. Calculer ω(1,2)(3,4)\omega_{(1,2)}(3,4).
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Solution by Ancient Tree

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  1. dxdx est l’application de R2\R^{2} dans R\R qui à tout (u,v)(u,v) associe uu. Autrement dit, c’est la projection sur chaque composante. De même, dydy est la projection sur la seconde composante.

On a donc ω(x,y)(u,v)=xu+yv\omega_{(x,y)}(u,v)=xu+yv.

  1. ω(1,2)(3,4)=1×3+2×4=11\omega_{(1,2)}(3,4)=1\times 3+2\times 4=11.
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