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In particolare + vale che: + \[ v \cdot w = \sum_{i=1}^n v_i w_i. \] + \end{definition} + + \begin{remark} + Dall'Algebra lineare, ogni iperpiano $P$ di $\RR^{n}$ è + rappresentabile tramite traslazione di una giacitura che è ortogonale rispetto a una retta, + ossia esistono sempre $c \in \RR$ e $v \in \RR^{n}$ tale per cui: + \[ x \in P \iff x \cdot v = c. \] + \end{remark} + + \begin{definition}[derivata direzionale] + Dati $x_0 \in \RR^n$, $f : \RR^n \to \RR$, e + $v \in \RR^n$, definisco la \textbf{derivata direzionale} + come: + \[ \frac{\partial f}{\partial v}(x_0) = \lim_{\eps \to 0} \frac{f(x + \eps v) - f(x)}{\eps}. \] + \end{definition} + + \begin{remark} + Si osserva che vale la seguente identità: + \[ \frac{\partial f}{\partial \lambda v} = \lambda \frac{\partial f}{\partial v}, \] + e che se $v = 0$, allora la derivata direzionale vale + sempre $0$. + \end{remark} + + \begin{remark} + Non vale la linearità sui vettori della derivata + direzionale, ossia, in generale, vale che: + \[ \frac{\partial f}{\partial (v + w)} \neq \frac{\partial f}{\partial v} + \frac{\partial f}{\partial w}. \] + Se infatti si definisce $f$ tale per cui: + \[ f(x, y) = \begin{cases} + \frac{x^2 y}{x^2 + y^2} & (x, y) \neq 0, + \\ 0 & (x, y) = 0, + \end{cases} \] + allora $\frac{\partial f}{\partial e_1}(0) = + \frac{\partial f}{\partial e_2}(0) = 0$, ma + $\frac{\partial f}{\partial (1,1)}(0) = \frac{1}{2}$. + \end{remark} + + \begin{remark} + Trovando un'analogia con $\RR$, vale la seguente identità: + \[ f(x_0 + \eps v) = f(x_0) + \eps \frac{\partial f}{\partial v}(x_0) + o(\abs{\eps v}). \] + In particolare si osserva che l'$o$-piccolo dipende dal + vettore direzionale scelto. + \end{remark} + + \begin{definition}[derivata parziale] + Si definisce \textbf{derivata parziale} rispetto a + $x_i$, la derivata direzionale rispetto al vettore + $e_i$, e si indica con: + \[ \frac{\partial f}{\partial x_i} := \frac{\partial f}{\partial e_i} \] + \end{definition} + + \begin{remark} + Se $\frac{\partial f}{\partial v}$ fosse lineare su $v$, + allora si potrebbe riscrivere la derivata direzionale come: + \[ \frac{\partial f}{\partial v} = \nabla \! f \, v, \quad + \nabla f = \left(\frac{\partial f}{\partial x_1} \cdots \frac{\partial f}{\partial x_n}\right), \] + dove $\nabla f$ è così composto perché in ogni colonna + raccoglie la sua valutazione nella base canonica, ossia + le derivate parziali. + \end{remark} + + \begin{definition}[gradiente di $f$] + Si definisce \textbf{gradiente} di una funzione $f : \RR^n \to \RR$ il vettore: + \[ \nabla f = \left(\frac{\partial f}{\partial x_1} \cdots \frac{\partial f}{\partial x_n}\right). \] + \end{definition} + + \begin{definition}[differenziabilità] + Si dice che $f$ è \textbf{differenziabile} se esiste + $\omega \in \RR^n$ tale per cui: + \[ f(x) = f(x_0) + (x-x_0) \cdot \omega + o(\abs{x-x_0}). \] + In tal caso si dice che $\omega$ è il suo \textbf{differenziale} e si indica con $Df(x_0)$. + \end{definition} +\end{document}