Diffgeo: Beispiele: Kugel

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Die Fläche

Geographische Koordinaten auf der Kugel

Parametrisiert wie in Geodätische Koordinatensysteme Geographische Koordinaten geschildert.

Parametrisierung

$\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}R\cdot cos(\lambda )\cdot sin(\varphi )\\ R\cdot sin(\lambda )\cdot sin(\varphi )\\ R\cdot cos(\varphi )\end{array}\right)$

$\varphi \in (0;\pi )\lambda \in [-\pi ;\pi )$

(Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

Tangentialraum :

$T_{S^2}(\lambda ,\varphi )=\{{\overrightarrow{x}}_{\lambda },{\overrightarrow{x}}_{\varphi }\}=\{\left(\begin{array}{c}-R\cdot \sin (\lambda )\cdot \sin (\varphi )\\ R\cdot \cos (\lambda )\cdot \sin (\varphi )\\ 0\end{array}\right),\left(\begin{array}{c}R\cdot \cos (\lambda )\cdot \cos (\varphi )\\ R\cdot \sin (\lambda )\cdot \cos (\varphi )\\ -R\cdot \sin (\varphi )\end{array}\right)\}$

Flächennormale

$N_{S^2}(\lambda ,\varphi )=\frac{({\overrightarrow{x}}_{\lambda }\wedge {\overrightarrow{x}}_{\varphi })}{|{\overrightarrow{x}}_{\lambda }\wedge {\overrightarrow{x}}_{\varphi }|}=\left(\begin{array}{c}-cos(\lambda )\cdot \sin (\varphi )\\ -sin(\lambda )\cdot \sin (\varphi )\\ -cos(\varphi )\end{array}\right)$

erste Fundamentalform

erste Fundamentalgrößen

Siehe hier:

$g_{11}=⟨{\overrightarrow{x}}_{\varphi },{\overrightarrow{x}}_{\varphi }⟩=R^2\sin {(\varphi )}^2$

$g_{12}=g_{21}=⟨{\overrightarrow{x}}_{\varphi },{\overrightarrow{x}}_{\lambda }⟩=0$

$g_{22}=⟨{\overrightarrow{x}}_{\lambda },{\overrightarrow{x}}_{\lambda }⟩=R^2$

erster Fundamentaltensor

$𝐆=\left(\begin{array}{cc}{g}_{11}& g_{12}\\ g_{21}& g_{22}\end{array}\right)=\left(\begin{array}{cc}{R}^2\sin {(\varphi )}^2& 0\\ 0& R^2\end{array}\right)$

Inverser erster Fundamentaltensor

${𝐆}^{-1}=\frac{1}{g_{11}{g}_{22}-(g_{12}{)}^2}\left(\begin{array}{cc}{g}_{22}& -g_{21}\\ -g_{12}& g_{11}\end{array}\right)=\left(\begin{array}{cc}{g}^{11}& g^{12}\\ g^{21}& g^{22}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{R^2\cdot {\sin }^2(\varphi )}& 0\\ 0& \frac{1}{R^2}\end{array}\right)$

zweite Fundamentalform

zweifache Ableitungen

${\overrightarrow{x}}_{\lambda \lambda }=\left(\begin{array}{c}-R\cdot \cos (\lambda )\cdot \sin (\varphi )\\ -R\cdot \sin (\lambda )\cdot \sin (\varphi )\\ 0\end{array}\right)$

${\overrightarrow{x}}_{\lambda \varphi }=\left(\begin{array}{c}-R\cdot \sin (\lambda )\cdot \cos (\varphi )\\ R\cdot \cos (\lambda )\cdot \cos (\varphi )\\ 0\end{array}\right)={\overrightarrow{x}}_{\varphi \lambda }$

${\overrightarrow{x}}_{\varphi \varphi }=\left(\begin{array}{c}-R\cdot \cos (\lambda )\cdot \sin (\varphi )\\ -R\cdot \sin (\lambda )\cdot \sin (\varphi )\\ -R\cdot \cos (\varphi )\end{array}\right)$

zweite Fundamentalgrößen

Hier nachschauen!

$b_{11}=⟨{\overrightarrow{x}}_{\varphi \varphi },N_{S^2}(\lambda ,\varphi )⟩=-R\cdot \sin {(\varphi )}^2$

$b_{12}=b_{21}=⟨{\overrightarrow{x}}_{\varphi \lambda },N_{S^2}(\lambda ,\varphi )⟩=0$

$b_{22}=⟨{\overrightarrow{x}}_{\lambda \lambda },N_{S^2}(\lambda ,\varphi )⟩=-R$

zweiter Fundamentaltensor

$𝐁=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)=\left(\begin{array}{cc}-R\cdot \sin {(\varphi )}^2& 0\\ 0& -R\end{array}\right)=\frac{-1}{R}\cdot 𝐆$

Krümmung

Definition

Hauptkrümungen

Bemerkung: Dies ist eine Variante, die jeweiligen Hauptkrümmungen mittels erster und zweiter Fundamentalform zu berechnen. Es führen aber viele Wege nach Rom.

$𝐀:=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$

so dass:

$-𝐆\cdot 𝐀=𝐁\iff 𝐀=-{𝐆}^{\mathbf{-}\mathrm{𝟏}}\cdot 𝐁$

$\Rightarrow 𝐀=-\left(\begin{array}{cc}\frac{1}{R^2\cdot {\sin }^2(\varphi )}& 0\\ 0& \frac{1}{R^2}\end{array}\right)\cdot \left(\begin{array}{cc}-R\cdot \sin {(\varphi )}^2& 0\\ 0& -R\end{array}\right)=\left(\begin{array}{cc}\frac{1}{R}& 0\\ 0& \frac{1}{R}\end{array}\right)$

Daraus lässt sich die Gaußsche und Mittlere Krümmung berechnen:

$K=a_{11}\cdot a_{22}=\frac{1}{R^2}$

$H=\frac{a_{11}+a_{22}}{2}=\frac{1}{R}$

Christoffelsymbole

Siehe hier. Mit u¹ =$\lambda .$, u² = $\varphi$ $\alpha =1$, $\beta =1$, $\gamma =1$

${\mathrm{\Gamma }}_{11}^1:=\frac{1}{2}{g}^{11}(\frac{\partial g_{11}}{\partial u^1}+\frac{\partial g_{11}}{\partial u^1}-\frac{\partial g_{11}}{\partial u^1})+\frac{1}{2}{g}^{12}(\frac{\partial g_{12}}{\partial u^1}+\frac{\partial g_{21}}{\partial u^1}-\frac{\partial g_{11}}{\partial u^2})=0$

$\alpha =2$, $\beta =1$, $\gamma =1$

${\mathrm{\Gamma }}_{12}^1:=\frac{1}{2}{g}^{11}(\frac{\partial g_{11}}{\partial u^2}+\frac{\partial g_{12}}{\partial u^1}-\frac{\partial g_{21}}{\partial u^1})+\frac{1}{2}{g}^{12}(\frac{\partial g_{12}}{\partial u^2}+\frac{\partial g_{22}}{\partial u^1}-\frac{\partial g_{21}}{\partial u^2})=\frac{\cos (\varphi )}{\sin (\varphi )}$

$\alpha =1$, $\beta =2$, $\gamma =1$

${\mathrm{\Gamma }}_{11}^2:=\frac{1}{2}{g}^{21}(\frac{\partial g_{11}}{\partial u^1}+\frac{\partial g_{11}}{\partial u^1}-\frac{\partial g_{11}}{\partial u^1})+\frac{1}{2}{g}^{22}(\frac{\partial g_{12}}{\partial u^1}+\frac{\partial g_{21}}{\partial u^1}-\frac{\partial g_{11}}{\partial u^2})=-\sin (\varphi )\cdot \cos (\varphi )$

$\alpha =1$, $\beta =1$, $\gamma =2$

${\mathrm{\Gamma }}_{21}^1={\mathrm{\Gamma }}_{12}^1$

$\alpha =2$, $\beta =1$, $\gamma =2$,

${\mathrm{\Gamma }}_{12}^2:=\frac{1}{2}{g}^{11}(\frac{\partial g_{21}}{\partial u^2}+\frac{\partial g_{12}}{\partial u^2}-\frac{\partial g_{22}}{\partial u^1})+\frac{1}{2}{g}^{12}(\frac{\partial g_{22}}{\partial u^2}+\frac{\partial g_{22}}{\partial u^2}-\frac{\partial g_{22}}{\partial u^2})=0$

$\alpha =1$, $\beta =2$, $\gamma =2$

${\mathrm{\Gamma }}_{21}^2={\mathrm{\Gamma }}_{12}^2$

$\alpha =2$, $\beta =2$, $\gamma =2$

${\mathrm{\Gamma }}_{22}^2:=\frac{1}{2}{g}^{21}(\frac{\partial g_{21}}{\partial u^2}+\frac{\partial g_{12}}{\partial u^2}-\frac{\partial g_{22}}{\partial u^1})+\frac{1}{2}{g}^{22}(\frac{\partial g_{22}}{\partial u^2}+\frac{\partial g_{22}}{\partial u^2}-\frac{\partial g_{22}}{\partial u^2})=0$

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