I have the following problem:
I need to compute $\int_M f(x) dS(x)$ where $f(x,y,z)=\sqrt{1+x^2+y^2}$ and $M=\{(x,y,z):z=xy, x^2+y^2\leq 1\}$
I wanted to use the formula for an integral over a submanifold, but then I don't know if I need to write $M$ as a union of disjoint sets $M_i$ or not. Furthermore I don't see how to find a chart in this situation. I thought using polar coordinates, but this somehow doesn't give me the same set.
In the meantime we found a chart, namely $$\phi(r,\theta)=(r\cos\theta, r\sin\theta,r^2\cos\theta \sin\theta)$$
Now I wanted to compute the gram determinant therefore I first need to compute the gram matrix: for that one I got \begin{align} \langle v,v\rangle &=1+r^2\sin^2(2\theta),\\ \langle v,w\rangle =\langle w,v\rangle &=\frac{r^3}{2}\cos(4\theta),\\ \langle w,w\rangle &=r^2+r^4\cos^2(2\theta) \end{align} but I'm not sure if this is correct because it seems to me really ugly
Could someone help me?