EDB — 1H1

Exercises

1. [1H1] In the same hypotheses of the exercise [1GZ]↺↻, we also assume that \(f∈ C^ 1(A)\).

   • We decompose \(y=(y',y_ n),∈ℝ^ n\) as we did for \(x\). We define the function \(G:V→ ℝ^ n\) as \(G(y)=(y',\tilde g(y))\). Let \(W=G(V)\) be the image of \(V\), show that \(W⊆ U\) and that \(W\) is open.

   • Show that is \(G:V→ W\) is a diffeomorphism; and that its inverse is the map \(F(x)=(x',f(x))\).

   • Let’s define \(\tilde f = f ◦ G\). Show that \(\tilde f(x)=x_ n\).

   (This exercise will be used, together with [1GB]↺↻, to address constrained problems, in Section [2D5]↺↻).

   Solution 1

   [1H2]↺↻