diff options
Diffstat (limited to 'ggs/n2.lyx')
| -rw-r--r-- | ggs/n2.lyx | 213 |
1 files changed, 169 insertions, 44 deletions
@@ -833,6 +833,128 @@ Así, está bien definido por lo anterior y cumple las propiedades. \end_layout +\begin_layout Standard + +\series bold +Lema de homogeneidad de las geodésicas: +\series default + Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + +, +\begin_inset Formula $\gamma_{v}:I_{v}\to S$ +\end_inset + + la geodésica maximal con condiciones iniciales +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $v$ +\end_inset + + y +\begin_inset Formula $\lambda\in\mathbb{R}^{*}$ +\end_inset + +, entonces +\begin_inset Formula $\gamma_{\lambda v}:I_{\gamma v}\to S$ +\end_inset + + viene dada por +\begin_inset Formula $I_{\lambda v}=\frac{1}{\lambda}I_{v}=\{\frac{t}{v}\}_{t\in I_{v}}$ +\end_inset + + y +\begin_inset Formula $\gamma_{\lambda v}(t)=\gamma_{v}(\lambda t)$ +\end_inset + + para todo +\begin_inset Formula $t\in I_{\lambda v}$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + con +\begin_inset Formula $\alpha(t):=\gamma_{v}(\lambda t)$ +\end_inset + +, claramente +\begin_inset Formula $I=\frac{1}{\lambda}I_{v}$ +\end_inset + +, pero +\begin_inset Formula $\alpha$ +\end_inset + + es una reparametrización afín de +\begin_inset Formula $\gamma$ +\end_inset + + y por tanto es una geodésica, +\begin_inset Formula $\alpha(0)=\gamma(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=\lambda\gamma'_{v}(0)=\lambda v$ +\end_inset + +, de modo que por unicidad es +\begin_inset Formula $\alpha\equiv\gamma_{\lambda v}|_{I}$ +\end_inset + +e +\begin_inset Formula $I=\frac{1}{\lambda}I_{v}\subseteq I_{\lambda v}$ +\end_inset + +. + Ahora bien, sea +\begin_inset Formula $w:=\lambda v$ +\end_inset + + y +\begin_inset Formula $\beta:I'\to S$ +\end_inset + + dada por +\begin_inset Formula $\beta(t):=\gamma_{w}(\frac{1}{\lambda}v)$ +\end_inset + +, por el mismo argumento es +\begin_inset Formula $I'=\lambda I_{w}=\lambda I_{\lambda v}\subseteq I_{v}$ +\end_inset + +, de modo que +\begin_inset Formula $I_{\lambda v}\subseteq\frac{1}{\lambda}I_{v}$ +\end_inset + + e +\begin_inset Formula $I_{\lambda v}=\frac{1}{\lambda}I_{v}$ +\end_inset + +, con +\begin_inset Formula $\alpha=\gamma_{\lambda v}$ +\end_inset + +. +\end_layout + \begin_layout Section Ecuaciones diferenciales lineales \end_layout @@ -949,34 +1071,31 @@ Hallar bases . \end_layout -\begin_layout Enumerate -Respecto de la base +\begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout -{ + +\backslash +eremember \end_layout \end_inset -\begin_inset Formula -\begin{align*} -{\cal B}:= & (w_{11},\dots,w_{1p_{1}},\dots,w_{r1},\dots,w_{rp_{r}},\\ - & \,v_{11},u_{11},\dots,v_{1q_{1}},u_{1q_{1}},\dots,v_{s1},u_{s1},\dots,v_{sq_{s}},u_{sq_{s}}), -\end{align*} - -\end_inset - +\end_layout +\begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout -} + +\backslash +sremember{EDO} \end_layout \end_inset @@ -984,44 +1103,43 @@ status open \end_layout -\begin_layout Standard -\begin_inset ERT +\begin_layout Enumerate +\begin_inset Argument item:1 status open \begin_layout Plain Layout - - -\backslash -eremember +3. \end_layout \end_inset - -\end_layout - -\begin_layout Standard +Respecto de la base \begin_inset ERT status open \begin_layout Plain Layout - -\backslash -sremember{EDO} +{ \end_layout \end_inset -\end_layout +\begin_inset Formula +\begin{align*} +{\cal B}:= & (w_{11},\dots,w_{1p_{1}},\dots,w_{r1},\dots,w_{rp_{r}},\\ + & \,v_{11},u_{11},\dots,v_{1q_{1}},u_{1q_{1}},\dots,v_{s1},u_{s1},\dots,v_{sq_{s}},u_{sq_{s}}), +\end{align*} -\begin_layout Enumerate -\begin_inset Argument item:1 +\end_inset + + +\begin_inset ERT status open \begin_layout Plain Layout +} \end_layout \end_inset @@ -1187,24 +1305,10 @@ status open \end_inset -Si -\begin_inset Formula $A$ -\end_inset - - es [de la primera forma][...], [...] -\begin_inset Formula -\[ -e^{tA}[...]=e^{t\lambda}\begin{pmatrix}1\\ -t & 1\\ -\frac{t^{2}}{2} & t & 1\\ -\vdots & \ddots & \ddots & \ddots\\ -\frac{t^{n-1}}{(n-1)!} & \cdots & \frac{t^{2}}{2} & t & 1 -\end{pmatrix} -\] - -\end_inset +\end_layout +\begin_layout Standard \begin_inset ERT status open @@ -1237,6 +1341,27 @@ sremember{EDO} \end_layout \begin_layout Standard +Si +\begin_inset Formula $A$ +\end_inset + + es [de la primera forma][...], [...] +\begin_inset Formula +\[ +e^{tA}[...]=e^{t\lambda}\begin{pmatrix}1\\ +t & 1\\ +\frac{t^{2}}{2} & t & 1\\ +\vdots & \ddots & \ddots & \ddots\\ +\frac{t^{n-1}}{(n-1)!} & \cdots & \frac{t^{2}}{2} & t & 1 +\end{pmatrix} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard Si [es de la segunda][...], \begin_inset Formula \[ |