The problem is about Gaussian Curvature When we define $M=r(u,v)$ be the surface parametrized by $(u,v)$ in $\mathbb{R}^3$ and $N$ be a unit normal vector of $M$. In my lecture we denote Gaussian curvature $\kappa=\det(B)/\det(A)$ where $$A=\begin{pmatrix} <r_u,r_u>&&<r_u,r_v>\\ <r_v,r_u>&&<r_v,r_v> \end{pmatrix} \qquad\text{and}\qquad B=\begin{pmatrix} <N_u,r_u>&&<N_u,r_v>\\ <N_v,r_u>&&<N_v,r_v> \end{pmatrix}.$$
Then show that
- $N_u,N_v$ can be written as linear combination of $r_u,r_v$
- if $(N_u,N_v)=(r_u,r_v)\begin{pmatrix} a&&b \\ c&&d \end{pmatrix}$ show that $\kappa=\det\begin{pmatrix} a&&b \\ c&&d \end{pmatrix}$
I found some useful answers in Calculating mean and Gaussian curvature and now try to solve the problem here but somehow still cannot solve it. Let $$M=r(u,v)=\begin{pmatrix}u\\v\\r(u,v)\end{pmatrix}\quad\text{ then }\quad r_u=\begin{pmatrix}1\\0\\r_u\end{pmatrix},\quad r_v=\begin{pmatrix}0\\1\\r_v\end{pmatrix}.$$ Therefore $$A=\begin{pmatrix}1+r_u^2&& r_ur_v\\r_vr_u&&1+r_v^2\end{pmatrix}$$ but when calculated unit normal vector I got $$N=\frac{1}{\sqrt{r_u^2+r_v^2+1}}\begin{pmatrix}-r_u\\-r_v\\1\end{pmatrix}$$ so $$N_u=\frac{1}{(r_u^2+r_v^2+1)^\frac{3}{2}}\begin{pmatrix}r_u(r_{uu}+r_{uv})-r_{uu}\\r_v(r_{uu}+r_{uv})-r_{uv}\\-r_{uu}-r_{uv}\end{pmatrix},\qquad N_v=\frac{1}{(r_u^2+r_v^2+1)^\frac{3}{2}}\begin{pmatrix}r_u(r_{vv}+r_{uv})-r_{uv}\\r_v(r_{vv}+r_{uv})-r_{vv}\\-r_{vv}-r_{uv}\end{pmatrix}$$ but I still cannot express $N_u,N_v$ in term of $r_u,r_v$ and find the $\kappa$. Did I misunderstand or miscalculate something?
