Abstract
Having acquired the language of vector fields, we return to differential equations and give a generalization of the local existence theorem known as the Frobenius theorem, whose proof will be reduced to the standard case discussed in Chapter IV. We state the theorem in ยง1. Readers should note that one needs only to know the definition of the bracket of two vector fields in order to understand the proof. It is convenient to insert also a formulation in terms of differential forms, for which the reader needs to know the local definition of the exterior derivative. However, the condition involving differential forms is proved to be equivalent to the vector field condition at the very beginning, and does not reappear explicitly afterwards.
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ยฉ 1999 Springer Science+Business Media New York
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Lang, S. (1999). The Theorem of Frobenius. In: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol 191. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0541-8_6
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DOI: https://doi.org/10.1007/978-1-4612-0541-8_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6810-9
Online ISBN: 978-1-4612-0541-8
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