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

$\overrightarrow{x}=v\cos u{\overrightarrow{e}}_1+v\sin u{\overrightarrow{e}}_2+A\cdot v^2{\overrightarrow{e}}_3$

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

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(Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

${\overrightarrow{g}}_1={\overrightarrow{x}}_u=-v\sin u\cdot {\overrightarrow{e}}_1+v\cos u\cdot {\overrightarrow{e}}_2$

${\overrightarrow{g}}_2={\overrightarrow{x}}_v=\cos u\cdot {\overrightarrow{e}}_1+\sin u\cdot {\overrightarrow{e}}_2+2Av\cdot {\overrightarrow{e}}_3$

${\overrightarrow{g}}_3=\frac{{\overrightarrow{x}}_u(u)\times {\overrightarrow{x}}_v(v)}{||{\overrightarrow{x}}_u(u)\times {\overrightarrow{x}}_v(v)||}=\frac{2Av\cos u}{\sqrt{4A^2{v}^2+1}}\cdot {\overrightarrow{e}}_1+\frac{2Av\sin u}{\sqrt{4A^2{v}^2+1}}\cdot {\overrightarrow{e}}_2-\frac{1}{\sqrt{4A^2{v}^2+1}}\cdot {\overrightarrow{e}}_3$

erste Fundamentalform

erste Fundamentalgrößen

Siehe hier:

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

$g_{12}=G_{21}={\overrightarrow{x}}_u\cdot {\overrightarrow{x}}_v=-v\sin u\cos u+v\cos u\sin u=0$

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

erster Fundamentaltensor

$𝐆=\left(\begin{array}{cc}{g}_{11}& g_{12}\\ g_{21}& g_{22}\end{array}\right)=\left(\begin{array}{cc}{v}^2& 0\\ 0& 1+4A^2{v}^2\end{array}\right)$

Inverser erster Fundamentaltensor

${𝐆}^{-1}=\left(\begin{array}{cc}{g}^{11}& g^{12}\\ g^{21}& g^{22}\end{array}\right)=\frac{1}{v^2+4A^2{v}^4}\left(\begin{array}{cc}1+4A^2{v}^2& 0\\ 0& v^2\end{array}\right)$

g₁₂ ist Null, die Parameterlinien stehen also senkrecht aufeinander.

zweite Fundamentalform

zweifache Ableitungen

${\overrightarrow{x}}_{uu}=-v\cos u\cdot {\overrightarrow{e}}_1-v\sin u\cdot {\overrightarrow{e}}_2$

${\overrightarrow{x}}_{uv}=-\sin u\cdot {\overrightarrow{e}}_1+\cos u\cdot {\overrightarrow{e}}_2$

${\overrightarrow{x}}_{vv}={\overrightarrow{x}}_{vu}=2A\cdot {\overrightarrow{e}}_3$

zweite Fundamentalgrößen

Hier nachschauen!

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

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

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

zweiter Fundamentaltensor

$𝐁=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)=\left(\begin{array}{cc}L& M\\ M& N\end{array}\right)=\left(\begin{array}{cc}\frac{-2A{v}^2}{\sqrt{4A^2{v}^2+1}}& 0\\ 0& \frac{-2A}{\sqrt{4A^2{v}^2+1}}\end{array}\right)$

Christoffelsymbole

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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