I had this exam question a few days ago and I still cannot figure out how to solve it:
Do note that I was taught differential geometry in a different language. If something doesn't make sense, point it out to me and I'll try to translate it better
Let the coefficients of the first fundamental form of a regular surface $X(u,v)$ be:
$$E=1+4u^2\;,\;F=\frac43uv\;,\; G=1+\frac43v^2$$
Find under which constant angle $\theta$ these two following curves intersect:
$c_1(t)=X(\cos(t),\sin(t),\; t\in(0,2\pi)$
$c_2(s)=X(s,\sqrt3s),\; s>0$
I know that $\cos(\phi)=\frac{F}{\sqrt{EG}}$, where $\phi$ is the angle between the two parametric lines(this is a possible mistranslation) of the surface, but that doesn't give me a lot of stuff to go on. I found the Gaussian Curvature to be $K_G=\frac{108}{\left(4 u^2 \left(8 v^2+9\right)+12 v^2+9\right)^2}$ using Brioschi's Formula but that didn't help either.
The curvature of the surface is not relevant. It's a strange question; ordinarily we ask if two surfaces intersect at a constant angle along a curve.
Here, we ask when the curves $c_1$ and $c_2$ intersect. So you need values of $t$ for which $\sin t = \sqrt3\cos t$. At such points, you want to find the angle(s) between the tangent vectors $c_1'(t)$ and $c_2'(s)$.
In terms of the parametrization, you have the tangent vectors $c_1'(t)=(-\sin t,\cos t)$ and $c_2'(s) = (1,\sqrt3)$. Use the first fundamental form and the fact that $\sin t = \sqrt3\cos t$ to find the angle between these two vectors.