gcs+

1 files changed, 331 insertions, 0 deletions

diff --git a/gcs/n3.lyx b/gcs/n3.lyx
index 61fab42..03db12a 100644
--- a/gcs/n3.lyx
+++ b/gcs/n3.lyx
@@ -1563,5 +1563,336 @@ identidad de polarización:
 
 \end_layout
 
+\begin_layout Section
+Curvas geodésica y normal
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $S$
+\end_inset
+
+ una superficie regular y 
+\begin_inset Formula $V:\mathbb{R}\to T_{p}S$
+\end_inset
+
+ diferenciable, llamamos 
+\series bold
+derivada covariante
+\series default
+ a 
+\begin_inset Formula 
+\[
+\frac{DV}{dt}(t):=\pi_{T_{p}S}V'(t),
+\]
+
+\end_inset
+
+la proyección de 
+\begin_inset Formula $V'(t)$
+\end_inset
+
+ en 
+\begin_inset Formula $T_{p}S$
+\end_inset
+
+.
+ Propiedades: Sean 
+\begin_inset Formula $V,W:\mathbb{R}\to T_{p}S$
+\end_inset
+
+ y 
+\begin_inset Formula $f:I\subseteq\mathbb{R}\to\mathbb{R}$
+\end_inset
+
+ diferenciables, siendo 
+\begin_inset Formula $I$
+\end_inset
+
+ un intervalo:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{D(fV)}{dt}=f'V+f\frac{DV}{dt}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $\pi:=\pi_{T_{p}S}$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{D(fV)}{dt}=\pi((fV)')=\pi(fV'+f'V)=f\pi V'+f'\pi V=f\frac{DV}{dt}+f'V$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\frac{D(V+W)}{dt}=\frac{DV}{dt}+\frac{DW}{dt}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\frac{D(V+W)}{dt}=\pi((V+W)')=\pi V'+\pi W'=\frac{DV}{dt}+\frac{DW}{dt}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt}W\rangle+\langle V,\frac{DW}{dt}\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{dV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$
+\end_inset
+
+, pero dada una base ortonormal 
+\begin_inset Formula $(v_{1},v_{2},v_{3})$
+\end_inset
+
+ con 
+\begin_inset Formula $T_{p}S=\text{span}\{v_{1},v_{2}\}$
+\end_inset
+
+, si 
+\begin_inset Formula $\frac{dV}{dt}(t)=\sum_{i}x_{i}v_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $W(t)=\sum_{i}y_{i}v_{i}$
+\end_inset
+
+, 
+\begin_inset Formula $\langle\frac{dV}{dt}(t),W(t)\rangle=\sum_{i=1}^{3}x_{i}y_{i}\overset{y_{3}=0}{=}x_{1}y_{1}+x_{2}y_{2}=\langle\pi_{T_{p}S}\frac{dV}{dt}(t),W(t)\rangle=\langle\frac{DV}{dt}(t),W(t)\rangle$
+\end_inset
+
+, y análogamente para 
+\begin_inset Formula $\langle V,\frac{dW}{dt}\rangle$
+\end_inset
+
+, luego 
+\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $S$
+\end_inset
+
+ una superficie regular orientada por 
+\begin_inset Formula $N$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha:I\to S$
+\end_inset
+
+ una curva, entonces 
+\begin_inset Formula $\alpha'(t)\in T_{\alpha(t)}S$
+\end_inset
+
+ para 
+\begin_inset Formula $t\in I$
+\end_inset
+
+, pero en general 
+\begin_inset Formula $\alpha''(t)\notin T_{\alpha(t)}S$
+\end_inset
+
+, aunque se escribe de forma única como la suma de una 
+\series bold
+aceleración tangencial
+\series default
+ o 
+\series bold
+intrínseca
+\series default
+ 
+\begin_inset Formula $\alpha''(t)^{\top}\in T_{\alpha(t)}S$
+\end_inset
+
+ y una 
+\series bold
+aceleración normal
+\series default
+ o 
+\series bold
+extrínseca
+\series default
+ 
+\begin_inset Formula $\alpha''(t)^{\bot}\in\text{span}\{N(\alpha(t))\}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\alpha''(t)^{\top}=\frac{D\alpha'}{dt}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\alpha''(t)=\frac{D\alpha'}{dt}(t)+\langle\alpha''(t),N(\alpha(t))\rangle N(\alpha(t)).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\alpha:I\to S$
+\end_inset
+
+ una curva parametrizada por longitud de arco, el 
+\series bold
+triedro de Darboux
+\series default
+ es la base ortonormal positivamente orientada 
+\begin_inset Formula $(\alpha'(s),J\alpha'(s):=\alpha'(s)\wedge N(\alpha(s)),N(\alpha(s))\rangle$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s)$
+\end_inset
+
+, donde 
+\begin_inset Formula $\kappa_{g}:=\langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$
+\end_inset
+
+, es la 
+\series bold
+curvatura geodésica
+\series default
+ de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, cuyo signo depende de 
+\begin_inset Formula $N$
+\end_inset
+
+.
+ En efecto, 
+\begin_inset Formula $\langle\frac{D\alpha'}{ds}(s),\alpha'(s)\rangle=\langle\alpha''(s)-\langle\alpha''(s),N(\alpha(s))\rangle N(\alpha(s)),\alpha'(s)\rangle=\langle\alpha''(s),\alpha'(s)\rangle-\langle\alpha''(s),N(\alpha(s))\rangle\langle N(\alpha(s)),\alpha'(s)\rangle=0$
+\end_inset
+
+, y 
+\begin_inset Formula $\kappa_{g}(s)=\langle\frac{D\alpha'}{ds}(s),J\alpha'(s)\rangle=\langle\alpha''(s),J\alpha'(s)\rangle$
+\end_inset
+
+, pero 
+\begin_inset Formula $J\alpha'(s)$
+\end_inset
+
+ puede ser un vector o su opuesto según lo sea 
+\begin_inset Formula $N$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dada una curva 
+\begin_inset Formula $\alpha:I\to S$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal II}_{\alpha(t)}(\alpha'(t))=\langle\alpha''(t),N(\alpha(t))\rangle$
+\end_inset
+
+.
+ En efecto, como 
+\begin_inset Formula $\alpha'(t)\in T_{\alpha(t)}S$
+\end_inset
+
+ para cada 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $\langle\alpha'(t),N(\alpha(t))\rangle=0$
+\end_inset
+
+ y, derivando, 
+\begin_inset Formula $\langle\alpha''(t),N(\alpha(t))\rangle+\langle\alpha'(t),(N\circ\alpha)'(t)\rangle=0$
+\end_inset
+
+, pero 
+\begin_inset Formula $(N\circ\alpha)'(t)=dN_{\alpha(t)}(\alpha'(t))$
+\end_inset
+
+, luego 
+\begin_inset Formula $\langle\alpha''(t),N(\alpha(t))\rangle=-\langle\alpha'(t),dN_{\alpha(t)}(\alpha'(t))\rangle=\langle\alpha'(t),A_{\alpha(t)}\alpha'(t)\rangle={\cal II}_{\alpha(t)}(\alpha'(t))$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Entonces, dados 
+\begin_inset Formula $p\in S$
+\end_inset
+
+ y 
+\begin_inset Formula $v\in T_{p}S$
+\end_inset
+
+ unitario, llamamos 
+\series bold
+curvatura normal
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+ en 
+\begin_inset Formula $p$
+\end_inset
+
+ en la dirección de 
+\begin_inset Formula $v$
+\end_inset
+
+ a 
+\begin_inset Formula $\kappa_{n}(v,p):={\cal II}_{p}(v)=\langle\alpha''(0),N(p)\rangle$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\alpha:(-\delta,\delta)\to S$
+\end_inset
+
+ una curva con 
+\begin_inset Formula $\alpha(0)=p$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha'(0)=v$
+\end_inset
+
+.
+\end_layout
+
 \end_body
 \end_document