在有限維度的流形上，有很多學者利用外微分式來探討變分學的問題。在此篇論文中，我們將以上的理論推廣至無窮維流形中，即建立希爾伯特流形上的外微分系統，並利用此理論來研究希爾伯特流形上的變分問題。

Calculus of variation on finite dimensional manifolds via exterior differential systems were expounded in the books of Sternberg, Bryant and Griffiths. Here we plan to extend the theory of exterior differential systems and study the applications to calculus of variation on Hilbert manifolds.

Chapter 1: Introduction 1
Chapter 2: Hilbert Manifold 3
2.1 Calculus on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Hilbert manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 3: Bundles 12
3.1 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 General construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 4: Alternating linear forms 17
Chapter 5: Exterior Differential systems 20
5.1 Exterior differential forms on a Hilbert space . . . . . . . . . . . . . . . . . 20
5.2 Exterior differential forms on a Hilbert manifold . . . . . . . . . . . . . . . . 22
5.3 Exterior differential systems on Hilbert manifold . . . . . . . . . . . . . . . 26
Chapter 6: Calculus of Variation 29
6.1 Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Euler-Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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