homework_sept23_2025.pdf

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Math 230a (Fall 2025) Yum-Tong Siu 1 Math 230a Homework # 3 Assigned September 23, 2025 due September 30, 2025 Problem 1. (Frobenius Integrabiltiy Proved by Integrating One Vector Field at a Time and Using Uniqueness of Ordinary Differential Equations) Let U be a nonempty connected open neighborhood of the origin 0 in Rn and let ξ1 , · · · , ξq be C ∞ vector fields on U which are R-linearly independent at every point of U . Assume that Frobenius integrability condition [ξj , ξk ] ∈ ⟨ξ1 ,…

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