EDB — 1H1

Exercises

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

   • We decompose $y=(y^{\prime },y_n),\in {ℝ}^n$ as we did for $x$. We define the function $G:V\to {ℝ}^n$ as $G(y)=(y^{\prime },\tilde{g}(y))$. Let $W=G(V)$ be the image of $V$, show that $W\subseteq U$ and that $W$ is open.

   • Show that is $G:V\to W$ is a diffeomorphism; and that its inverse is the map $F(x)=(x^{\prime },f(x))$.

   • 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]↺↻