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Nel caso $n > 1$, coincide con il vettore $({f_i}'(t))_i)$. La notazione può essere iterata per ottenere le derivate successive. \item $\partial_{x_i} f(\vec{x})$, $\pd{f}{x_i}(\vec{x})$, $f_{x_i}(\vec{x})$ -- derivata parziale nella $i$-esima coordinata di una funzione $f : \RR^n \to \RR$ nel punto $\vec{x}$. La notazione può essere iterata per ottenere le diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex index 3136940..7a790c4 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/1-curve.tex @@ -6,7 +6,7 @@ Qualora non specificato, assumeremo l'utilizzo di funzioni di classe $C^\infty$. \smallskip Se $\vec{x} \in \RR^3$ e $f$ è una funzione relativa a una curva $\alpha$, ammettiamo l'abuso di notazione - $f(\vec{x})$, intendendo $f(\alpha\inv(\vec{x}))$ (e.g., $\kappa(P)$ per intendere $\kappa(\alpha\inv(P)))$). + $f(\vec{x})$, intendendo $f(\alpha\inv(\vec{x}))$; per esempio useremo $\kappa(P)$ per intendere $\kappa(\alpha\inv(P))$. \section{Definizioni preliminari} @@ -144,7 +144,7 @@ \begin{definition}[Curvatura] Sia $\beta$ una curva p.l.a., allora si definisce la \textbf{curvatura} $\kappa_\beta(s)$ di $\beta$ al tempo $s$ come - $\norm{T_\beta'(s)}$. \smallskip + $\norm{\dot{T_\beta}(s)}$. \smallskip Laddove è chiaro dal contesto quale sia $\beta$, scriviamo solo $\kappa(s)$. \end{definition} @@ -161,7 +161,7 @@ a ogni tempo $s$ il \textbf{versore normale} $N_\beta(s)$ così definito: \[ - N_\beta(s) \defeq \frac{T_\beta(s)}{\norm{T_\beta(s)}}. + N_\beta(s) \defeq \frac{\dot{T_\beta}(s)}{\norm{\dot{T_\beta}(s)}}. \] \end{definition}