91影视

The equation of smooth parametrization for a 鈥済eneralized cylinder鈥� S, given by extending the curve y = x^2.

$$\begin{gathered} n=r_v脳r_v \\ \begin{array}{r}\mathbf{n}={\mathbf{r}}_u脳{\mathbf{r}}_v\\ =鉄�1,2u,0鉄�脳鉄�0,0,1鉄�\\ =\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 2u& 0\\ 0& 0& 1\end{array}\right|\\ =[2u鈰�1鈭�0鈰�0]\mathbf{i}鈭�[1鈰�1鈭�0鈰�0]\mathbf{j}+[1鈰�0鈭�2u鈰�0]\mathbf{k}\\ =2u\mathbf{i}鈭�1\mathbf{j}+0\mathbf{k}\\ =2u\mathbf{i}鈭�\mathbf{j}.\end{array} \\ \mathrm{Therefore},\mathrm{the}\mathrm{required}\mathrm{normal}\mathrm{vector}\mathrm{for}\mathrm{the}\mathrm{above}\mathrm{cylinder}\mathrm{is}\mathrm{n}==2\mathrm{u}\mathbf{i}鈭�\mathbf{j}. \end{gathered}$$