EDB — 1G8

[1G8] Let \(W⊆ ℝ^ n\) be an open nonempty set, fix \(\overline x∈ W\). Let then \(𝜓:W→ℝ\) of class \(C^ 2\). Let \(∇𝜓(\overline x)\) be the row vector of coordinates \(\frac{\partial ~ }{\partial {x_ k}} 𝜓(\overline x)\) (which is the gradient of \(𝜓\), a special case of the ”Jacobian matrix”); we abbreviate it to \(D=∇𝜓(\overline x)\) for simplicity; let \(H\) be the Hessian matrix of components \(H_{h,k}= \frac{\partial {}^ 2}{\partial {x_ k x_ h}}𝜓 (\overline x) \); show the validity of Taylor’s formula at the second order

(note that the product \(D v\) is a matrix \(1× 1\) that we identify with a real number, and similarly for \(v^ t H v\)).

[ [1G9]↺↻]