diff options
Diffstat (limited to 'gcs')
| -rw-r--r-- | gcs/n1.lyx | 5 | ||||
| -rw-r--r-- | gcs/n3.lyx | 90 |
2 files changed, 65 insertions, 30 deletions
@@ -5,6 +5,9 @@ \save_transient_properties true \origin unavailable \textclass book +\begin_preamble +\input{../defs} +\end_preamble \use_default_options true \maintain_unincluded_children false \language spanish @@ -2037,7 +2040,7 @@ Curvas en el espacio \end_layout \begin_layout Standard -Sean +Sea \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset @@ -1573,17 +1573,54 @@ Sean \end_inset una superficie regular y -\begin_inset Formula $V:\mathbb{R}\to T_{p}S$ +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva regular, un +\series bold +campo de vectores a lo largo de +\begin_inset Formula $\alpha$ +\end_inset + + +\series default + es una función +\begin_inset Formula $V:I\to\mathbb{R}^{3}$ \end_inset - diferenciable, llamamos +, y es +\series bold +tangente +\series default + a +\begin_inset Formula $S$ +\end_inset + + (a lo largo de +\begin_inset Formula $\alpha$ +\end_inset + +) si para +\begin_inset Formula $t\in S$ +\end_inset + + es +\begin_inset Formula $V(t)\in T_{\alpha(t)}S$ +\end_inset + +. + Sea +\begin_inset Formula $V:I\to\mathbb{R}^{3}$ +\end_inset + + un campo de vectores tangente y diferenciable, llamamos \series bold derivada covariante \series default a \begin_inset Formula \[ -\frac{DV}{dt}(t):=\pi_{T_{p}S}V'(t), +\frac{DV}{dt}(t):=\pi_{T_{\alpha(t)}S}V'(t), \] \end_inset @@ -1598,11 +1635,11 @@ la proyección de . Propiedades: Sean -\begin_inset Formula $V,W:\mathbb{R}\to T_{p}S$ +\begin_inset Formula $V,W:I\to T_{p}S$ \end_inset y -\begin_inset Formula $f:I\subseteq\mathbb{R}\to\mathbb{R}$ +\begin_inset Formula $f:I\to\mathbb{R}$ \end_inset diferenciables, siendo @@ -1622,11 +1659,11 @@ la proyección de \begin_deeper \begin_layout Standard Si -\begin_inset Formula $\pi:=\pi_{T_{p}S}$ +\begin_inset Formula $\pi:=\pi_{T_{\alpha(t)}S}$ \end_inset , -\begin_inset Formula $\frac{D(fV)}{dt}=\pi((fV)')=\pi(fV'+f'V)=f\pi V'+f'\pi V=f\frac{DV}{dt}+f'V$ +\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 . @@ -1642,7 +1679,7 @@ Si \begin_deeper \begin_layout Standard -\begin_inset Formula $\frac{D(V+W)}{dt}=\pi((V+W)')=\pi V'+\pi W'=\frac{DV}{dt}+\frac{DW}{dt}$ +\begin_inset Formula $\frac{D(V+W)}{dt}=\pi((V+W)')=\pi(V')+\pi(W')=\frac{DV}{dt}+\frac{DW}{dt}$ \end_inset . @@ -1678,7 +1715,7 @@ Si \end_inset , -\begin_inset Formula $\langle\frac{dV}{dt}(t),W(t)\rangle=\sum_{i=1}^{3}x_{i}y_{i}\overset{y_{3}=0}{=}x_{1}y_{1}+x_{2}y_{2}=\langle\pi_{T_{p}S}\frac{dV}{dt}(t),W(t)\rangle=\langle\frac{DV}{dt}(t),W(t)\rangle$ +\begin_inset Formula $\langle\frac{dV}{dt}(t),W(t)\rangle=\sum_{i=1}^{3}x_{i}y_{i}\overset{y_{3}=0}{=}x_{1}y_{1}+x_{2}y_{2}=\langle\pi(\frac{dV}{dt}(t)),W(t)\rangle=\langle\frac{DV}{dt}(t),W(t)\rangle$ \end_inset , y análogamente para @@ -1686,7 +1723,7 @@ Si \end_inset , luego -\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$ +\begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{dV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{DW}{dt}\rangle$ \end_inset . @@ -1773,10 +1810,14 @@ triedro de Darboux . Entonces -\begin_inset Formula $\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s)$ +\begin_inset Formula +\[ +\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s), +\] + \end_inset -, donde + donde \begin_inset Formula $\kappa_{g}:=\langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$ \end_inset @@ -1794,15 +1835,10 @@ curvatura geodésica . En efecto, -\begin_inset Formula -\begin{multline*} -\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{multline*} - +\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 +, 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 @@ -2206,10 +2242,6 @@ direcciones principales . \end_layout -\begin_layout Standard -Ejemplos: -\end_layout - \begin_layout Enumerate Todas las direcciones del plano y la esfera son principales. \end_layout @@ -3135,15 +3167,15 @@ respecto de la base , entonces \begin_inset Formula -\begin{align*} +\[ \begin{pmatrix}-e & -f\\ -f & -g -\end{pmatrix} & =\begin{pmatrix}a_{11} & a_{21}\\ +\end{pmatrix}=\begin{pmatrix}a_{11} & a_{21}\\ a_{12} & a_{22} \end{pmatrix}\begin{pmatrix}E & F\\ F & G \end{pmatrix} -\end{align*} +\] \end_inset @@ -3888,7 +3920,7 @@ símbolos de Christoffel \end_inset y -\begin_inset Formula $\Gamma_{12}^{2}=\Gamma_{22}^{2}$ +\begin_inset Formula $\Gamma_{12}^{2}=\Gamma_{21}^{2}$ \end_inset , pues @@ -4103,9 +4135,9 @@ Como nos da \begin_inset Formula -\begin{multline*} +\[ \Gamma_{11}^{1}f+\Gamma_{11}^{2}g-\Gamma_{12}^{1}e-\Gamma_{12}^{1}f+e_{v}-f_{u}=0, -\end{multline*} +\] \end_inset |