So I have given a set $M=\{(x,y,z)\in\mathbb{R^3}:\frac{x^2}{R^2}+\frac{y^2}{r^2}+\frac{z^2}{r^2}=1\}$ where $0<r<R$. and I have to find the Gaussian curvature, which we do using a formula with determinants of two matrices, in the Point P=(R,0,0). The problem is that before I do anything I have to find a parametrization $\phi(u_{1},u_{2})$.
I tried to do find the parametrization, expressing $z$ depending from $x$ and $y$ and I got $\phi(u_{1},u_{2})=(u_{1},u_{2},\sqrt{\frac{-r^2u_{1}^2}{R^2}-u_{2}^2+r^2})$ (please correct me if this is wrong).
Now the following steps are to calculate the derivatives of this $\phi$ according to $u_{1}$ and $u_{2}$ and a lot of other computations until i come to the Gauss curvature. Calculating all of the steps with the square root term is not pretty and is very time consuming. In the cases where the point $P$ is already given, our professor does something which facilitates the process a lot. He kinda substitutes the point from early on so the terms get a lot more easier to work with. I see my notes but I don't understand this process. The main confusing thing is that the point P has 3 coordinates while $\phi$ takes two components, so I don't see where we substitute the point.
I have an exam coming up so I would be extremely thankful for some help and insight.
Thanks in advance for the help
PS. I am sorry for possible language or terminology errors but not english speaker and I don't study math in english.