Eingebetteter Torus/Einheitsnormalenfeld/Aufgabe/Lösung

Es ist

{}h(x,y,z)=x^{2}+y^{2}-2{\sqrt {x^{2}+y^{2}}}R+z^{2}+R^{2}\,

und ${}T=h^{-1}(r^{2})$ . Der Gradient von ${}h$ ist

{}\operatorname {Grad} \,h=2{\begin{pmatrix}x-xR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\\y-yR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\\z\end{pmatrix}}\,.

{}N(x,y,z)={\frac {1}{\sqrt {{\left(x-xR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\right)}^{2}+{\left(y-yR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\right)}^{2}+z^{2}}}}{\begin{pmatrix}x-xR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\\y-yR{\left(x^{2}+y^{2}\right)}^{-{\frac {1}{2}}}\\z\end{pmatrix}}\,.