EDB — 1GB

[1GB] Prerequisites:[1G8]↺↻.Let $V,W\subseteq {ℝ}^n$ be open nonempty sets, and $G:V\to W$ of class $C^2$. Fix $\bar{y}\in V$ and $\bar{x}=G(\bar{y})\in W$. Suppose that $𝜓:W\to ℝ$ is of class $C^2$; define $\widetilde{𝜓}=𝜓◦G$, then compare Taylor’s second-order formulas for $𝜓$ and $\widetilde{𝜓}$ (centered in $\bar{x}$ and $\bar{y}$, respectively). Assuming also that $G$ is a diffeomorphism, verify that

• $\bar{x}$ is a stationary point for $𝜓$ if and only if $\bar{y}$ is stationary point for $\widetilde{𝜓}$,

• and in this case the Hessians of $𝜓$ and $\widetilde{𝜓}$ are similar (i.e. the matrices are equal, up to coordinate changes).