# Diffgeo: Beispiele: Kugel

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# Die Fläche

Parametrisiert wie in Geodätische Koordinatensysteme Geographische Koordinaten geschildert.

# Parametrisierung

${\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}R\cdot cos{(\lambda )}\cdot sin{(\phi )}\\R\cdot sin{(\lambda )}\cdot sin{(\phi )}\\R\cdot cos{(\phi )}\end{pmatrix}}}$
${\displaystyle \phi \in (0;\pi )\,\lambda \in [-\pi ;\pi )}$

# (Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

Tangentialraum :

${\displaystyle T_{S^{2}}(\lambda ,\phi )=\lbrace {\vec {x}}_{\lambda },{\vec {x}}_{\phi }\rbrace ={\bigg \lbrace }{\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}},{\begin{pmatrix}R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\-R\cdot \sin {(\phi )}\end{pmatrix}}{\bigg \rbrace }}$

Flächennormale

${\displaystyle N_{S^{2}}(\lambda ,\phi )={\frac {({\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi })}{\vert {\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi }\vert }}={\begin{pmatrix}-cos{(\lambda )}\cdot \sin {(\phi )}\\-sin{(\lambda )}\cdot \sin {(\phi )}\\-cos{(\phi )}\end{pmatrix}}}$

# erste Fundamentalform

## erste Fundamentalgrößen

Siehe hier:

${\displaystyle g_{11}=\langle {\vec {x}}_{\phi },{\vec {x}}_{\phi }\rangle =R^{2}\sin {(\phi )}^{2}}$
${\displaystyle g_{12}=g_{21}=\langle {\vec {x}}_{\phi },{\vec {x}}_{\lambda }\rangle =0}$
${\displaystyle g_{22}=\langle {\vec {x}}_{\lambda },{\vec {x}}_{\lambda }\rangle =R^{2}}$

## erster Fundamentaltensor

${\displaystyle \mathbf {G} ={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}R^{2}\sin {(\phi )}^{2}&0\\0&R^{2}\end{pmatrix}}}$

## Inverser erster Fundamentaltensor

${\displaystyle \mathbf {G} ^{-1}={\frac {1}{g_{11}g_{22}-(g_{12})^{2}}}{\begin{pmatrix}g_{22}&-g_{21}\\-g_{12}&g_{11}\end{pmatrix}}={\begin{pmatrix}g^{11}&g^{12}\\g^{21}&g^{22}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{R^{2}\cdot \sin ^{2}{(\phi )}}}&0\\0&{\frac {1}{R^{2}}}\end{pmatrix}}}$

# zweite Fundamentalform

## zweifache Ableitungen

${\displaystyle {\vec {x}}_{\lambda \lambda }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}}}$
${\displaystyle {\vec {x}}_{\lambda \phi }={\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\0\end{pmatrix}}={\vec {x}}_{\phi \lambda }}$
${\displaystyle {\vec {x}}_{\phi \phi }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \cos {(\phi )}\end{pmatrix}}}$

## zweite Fundamentalgrößen

Hier nachschauen!

${\displaystyle b_{11}=\langle {\vec {x}}_{\phi \phi },N_{S^{2}}(\lambda ,\phi )\rangle =-R\cdot \sin {(\phi )}^{2}}$
${\displaystyle b_{12}=b_{21}=\langle {\vec {x}}_{\phi \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =0}$
${\displaystyle b_{22}=\langle {\vec {x}}_{\lambda \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =-R}$

## zweiter Fundamentaltensor

${\displaystyle \mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\frac {-1}{R}}\cdot \mathbf {G} }$

# Krümmung

## Hauptkrümungen

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

${\displaystyle \mathbf {A} :={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}$

so dass:

${\displaystyle -\mathbf {G} \cdot \mathbf {A} =\mathbf {B} \Leftrightarrow \mathbf {A} =-\mathbf {G^{-1}} \cdot \mathbf {B} }$
${\displaystyle \Rightarrow \mathbf {A} =-{\begin{pmatrix}{\frac {1}{R^{2}\cdot \sin ^{2}{(\phi )}}}&0\\0&{\frac {1}{R^{2}}}\end{pmatrix}}\cdot {\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\begin{pmatrix}{\frac {1}{R}}&0\\0&{\frac {1}{R}}\end{pmatrix}}}$

Daraus lässt sich die Gaußsche und Mittlere Krümmung berechnen:

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

# Christoffelsymbole

Siehe hier. Mit u1 =${\displaystyle \lambda .}$, u2 = ${\displaystyle \phi }$ ${\displaystyle \alpha =1}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =1}$

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =1}$

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =1}$

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =2}$

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =2}$,

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =2}$

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =2}$

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

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