Aloha :)$$\vec r(u_1,u_2,u_3)=\left(\begin{array}{c}\frac{1}{2}(u_1^2-u_2^2)\\u_1u_2+u_3^2\\u_3\end{array}\right)$$Die partiellen Ableitungen sind:
$$\frac{\partial\vec r}{\partial u_1}=\left(\begin{array}{c}u_1\\u_2\\0\end{array}\right)\quad;\quad\frac{\partial\vec r}{\partial u_2}=\left(\begin{array}{c}-u_2\\u_1\\0\end{array}\right)\quad;\quad\frac{\partial\vec r}{\partial u_3}=\left(\begin{array}{c}0\\2u_3\\1\end{array}\right)$$Damit lauten die Elemente des metrischen Tensors:
$$g_{11}=\frac{\partial\vec r}{\partial u_1}\cdot\frac{\partial\vec r}{\partial u_1}=\left(\begin{array}{c}u_1\\u_2\\0\end{array}\right)\left(\begin{array}{c}u_1\\u_2\\0\end{array}\right)=u_1^2+u_2^2$$$$g_{12}=\frac{\partial\vec r}{\partial u_1}\cdot\frac{\partial\vec r}{\partial u_2}=\left(\begin{array}{c}u_1\\u_2\\0\end{array}\right)\left(\begin{array}{c}-u_2\\u_1\\0\end{array}\right)=0$$$$g_{13}=\frac{\partial\vec r}{\partial u_1}\cdot\frac{\partial\vec r}{\partial u_3}=\left(\begin{array}{c}u_1\\u_2\\0\end{array}\right)\left(\begin{array}{c}0\\2u_3\\1\end{array}\right)=2u_2u_3$$$$g_{22}=\frac{\partial\vec r}{\partial u_2}\cdot\frac{\partial\vec r}{\partial u_2}=\left(\begin{array}{c}-u_2\\u_1\\0\end{array}\right)\left(\begin{array}{c}-u_2\\u_1\\0\end{array}\right)=u_1^2+u_2^2$$$$g_{23}=\frac{\partial\vec r}{\partial u_2}\cdot\frac{\partial\vec r}{\partial u_3}=\left(\begin{array}{c}-u_2\\u_1\\0\end{array}\right)\left(\begin{array}{c}0\\2u_3\\1\end{array}\right)=2u_1u_3$$$$g_{33}=\frac{\partial\vec r}{\partial u_3}\cdot\frac{\partial\vec r}{\partial u_3}=\left(\begin{array}{c}0\\2u_3\\1\end{array}\right)\left(\begin{array}{c}0\\2u_3\\1\end{array}\right)=4u_3^2+1$$
Auf Grund seiner Definition ist der metrische Tensor symmetrisch:$$\mathbf{g}=\left(\begin{array}{c}u_1^2+u_2^2 & 0 & 2u_2u_3\\0 & u_1^2+u_2^2 & 2u_1u_3\\ 2u_2u_3 & 2u_1u_3 &4u_3^2+1 \end{array}\right)$$
