Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.
Please do the following additional exercises.
- A*. Let F : R2 → R2 be given by
x = u2 - v2, y = 2 u v;
Find an open set U ⊆ R2 such that (3,4) ∈ U and V = F(U) is an open set, and find a C1 function G : V → U such that
G o F (u,v) = (u,v) for all (u,v)∈U andF o G (x,y) = (x,y) for all (x,y)∈V.
Find the differential dG(F(3,4)).
(G is a local inverse. Solve for G and check its properties. Do not use the Inverse Function Theorem, which guarantees the existence of local inverse near (3,4) assuming F is continuously differentiable near (3,4) and dF(3,4) is invertible.)