Diffgeo: Beispiele: Drehparaboloid

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Die Fläche

Ein Drehparaboloid entsteht durch Rotation einer in einer Ebene liegenden Parabel um eine Achse.

Zum Beispiel wenn die in der x-z liegende Parabel z=x² (Definitionsbereich begrenzt) um die z-Achse rotiert. Durch die Rotation ist der Rand der Fläche ein Kreis.

Parametrisierung

${\vec {x}}=v\cos u{\vec {e}}_{1}+v\sin u{\vec {e}}_{2}+A\cdot v^{2}{\vec {e}}_{3}$

$v\in [0,2\pi ],u\in [-\pi ,\pi ],A\in \mathbb {R}$

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(Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

${\vec {g}}_{1}={\vec {x}}_{u}=-v\sin u\cdot {\vec {e}}_{1}+v\cos u\cdot {\vec {e}}_{2}$

${\vec {g}}_{2}={\vec {x}}_{v}=\cos u\cdot {\vec {e}}_{1}+\sin u\cdot {\vec {e}}_{2}+2Av\cdot {\vec {e}}_{3}$

${\vec {g}}_{3}={\frac {{\vec {x}}_{u}(u)\times {\vec {x}}_{v}(v)}{||{\vec {x}}_{u}(u)\times {\vec {x}}_{v}(v)||}}={\frac {2Av\cos u}{\sqrt {4A^{2}v^{2}+1}}}\cdot {\vec {e}}_{1}+{\frac {2Av\sin u}{\sqrt {4A^{2}v^{2}+1}}}\cdot {\vec {e}}_{2}-{\frac {1}{\sqrt {4A^{2}v^{2}+1}}}\cdot {\vec {e}}_{3}$

erste Fundamentalform

erste Fundamentalgrößen

Siehe hier:

$g_{11}={\vec {x}}_{u}\cdot {\vec {x}}_{u}={\vec {x}}_{u}=v^{2}\sin ^{2}u+v^{2}\cos ^{2}u=v^{2}$

$g_{12}=G_{21}={\vec {x}}_{u}\cdot {\vec {x}}_{v}=-v\sin u\cos u+v\cos u\sin u=0$

$g_{22}={\vec {x}}_{v}\cdot {\vec {x}}_{v}=\cos ^{2}u+\sin ^{2}u+4A^{2}v^{2}$

erster Fundamentaltensor

$\mathbf {G} ={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}v^{2}&0\\0&1+4A^{2}v^{2}\end{pmatrix}}$

Inverser erster Fundamentaltensor

$\mathbf {G} ^{-1}={\begin{pmatrix}g^{11}&g^{12}\\g^{21}&g^{22}\end{pmatrix}}={\frac {1}{v^{2}+4A^{2}v^{4}}}{\begin{pmatrix}1+4A^{2}v^{2}&0\\0&v^{2}\end{pmatrix}}$

g₁₂ ist Null, die Parameterlinien stehen also senkrecht aufeinander.

zweite Fundamentalform

zweifache Ableitungen

${\vec {x}}_{uu}=-v\cos u\cdot {\vec {e}}_{1}-v\sin u\cdot {\vec {e}}_{2}$

${\vec {x}}_{uv}=-\sin u\cdot {\vec {e}}_{1}+\cos u\cdot {\vec {e}}_{2}$

${\vec {x}}_{vv}={\vec {x}}_{vu}=2A\cdot {\vec {e}}_{3}$

zweite Fundamentalgrößen

Hier nachschauen!

$b_{11}={\vec {x}}_{uu}\cdot {\vec {n}}={\frac {-2Av^{2}}{\sqrt {4A^{2}v^{2}+1}}}$

$b_{12}=b_{21}={\vec {x}}_{uv}\cdot {\vec {n}}=0$

$b_{21}={\vec {x}}_{vv}\cdot {\vec {n}}={\frac {-2A}{\sqrt {4A^{2}v^{2}+1}}}$

zweiter Fundamentaltensor

$\mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}L&M\\M&N\end{pmatrix}}={\begin{pmatrix}{\frac {-2Av^{2}}{\sqrt {4A^{2}v^{2}+1}}}&0\\0&{\frac {-2A}{\sqrt {4A^{2}v^{2}+1}}}\end{pmatrix}}$

Christoffelsymbole

Siehe hier. Mit u¹ = u, u² = v. $\alpha =1$, $\beta =1$, $\gamma =1$

$\Gamma _{11}^{1}:={\frac {1}{2}}g^{11}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})={\frac {1}{2}}{\frac {1}{v^{4}+4A^{2}v^{6}}}\cdot (0+0+0)+{\frac {1}{2}}\cdot 0=0$

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

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

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

$\Gamma _{11}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})=0+{\frac {1}{2(1+4A^{2}v^{2})}}\cdot (-2v)=-{\frac {v}{1+4A^{2}v^{2}}}$

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

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

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

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

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

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

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

$\Gamma _{22}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{21}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{22}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{2}}})={\frac {1}{2(1+4A^{2}v^{2})}}\cdot 8A^{2}v={\frac {4A^{2}}{1+4A^{2}v^{2}}}$

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