Course Queries Syllabus Queries 2 years ago
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I proof the global version of the inverse function theorem using the following local version:
Inverse function theorem [local]. Let E⊂Rn open, and let f:E→Rn be a C1 map. Let a∈E be such that det(f′(a))≠0. Then there exists open balls U∋a and V∋f(a) such that f:U→V is a C1 diffeomorphism.
I will proof:
Inverse function theorem [global] Let E⊂Rn open, and let f:E→Rn be an injective
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Best Answer
2 years ago
In our syllabus, we proved the local inverse function theorem: Let f:E→Rnf:E→Rn with E⊆RnE⊆Rn open be C1C1. If a⃗ ∈Ea→∈E with detf′(a⃗ )≠0detf′(a→)≠0, then there exist open sets U∋a⃗ U∋a→ and V∋f(a⃗ )V∋f(a→) such that f:U→Vf:U→V is a C1C1-diffeomorphism.
I have two questions which involve this theorem.
First, I am asked to prove the following 'global' version of the inverse function theorem, which states: Let f:E→font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: normal; font-family: