I may simply be overwhelmed by all the terms in this question, but I am at a point where I feel stuck:
Given a surface $X(u,v)$ with $u=u(t)$ and $v=v(t)$, and $F=0$, find a formula for the geodesic curvature in terms of the coefficients $E$,$F$,$G$, and the derivatives of $u(t)$ and $v(t)$.
So it is given that $S=X(u(t),v(t))$ and I know that $k_g = \vert \dot x,\ddot x,e_3 \vert$. The goal is to get $k_g$ in terms of the first fundamental form.
Here's what I have so far:
$$e_3=\frac{X_u \times X_v}{\Vert X_u \times X_v \Vert}$$
Which implies $$e_3=\frac{X_u \times X_v}{\sqrt{EG}}$$
I know that I am supposed to use the following:
$$X_{uu}=\Gamma{_1}^1{_1}e_1+\Gamma{_1}^2{_1}e_2+Le_3$$ $$X_{uv}=\Gamma{_1}^1{_2}e_1+\Gamma{_1}^2{_2}e_2+Me_3$$ $$X_{vv}=\Gamma{_2}^1{_2}e_1+\Gamma{_2}^2{_2}e_2+Ne_3$$
$$X_{uu}=\frac{GE_u}{2EG}e_1+\frac{-EE_v}{2EG}e_2+Le_3$$ $$X_{uv}=\frac{GE_v}{2EG}e_1+\frac{EG_u}{2EG}e_2+Me_3$$ $$X_{vv}=\frac{-GG_u}{2EG}e_1+\frac{EG_v}{2EG}e_2+Ne_3$$
And it's at this point that I feel like I don't know which direction to go. I've been staring at the problem for over an hour now and I am most confused with trying to rewrite $\dot x$ and $\ddot x$ in terms of $t$...which is how I think I should start.
Thank you for any direction.