Suppose that V is a bounded, nonempty, open set in Rn and that V Rn is 1 1 and continuously differentiable on V with 0 on V Let W Wj jN be an open covering of V and j jN be a Cp partition of unity on V subordinate to W, where p 1 Prove that ,j o 1 jN is a C1 partition of unity on (V) subordinate to...

Fix j N By the Inverse Function Theorem 1 hence also j 1 is C 1 on V In pa ...

Suppose that V is a bounded, nonempty, open set in Rn and that : V Rn

Question:

Suppose that V is a bounded, nonempty, open set in Rn and that ɸ: V → Rn is 1 - 1 and continuously differentiable on V with Δɸ ≠ 0 on V. Let W = {Wj}j∈N be an open covering of V and {ɸj}j∈N be a Cp partition of unity on V subordinate to W, where p > 1. Prove that {∅,j o ∅-1}j∈N is a C1 partition of unity on ɸ(V) subordinate to the open covering {ɸ(Wj)}j∈N.