My end goal is to show why Gauss's original equation for curvature is equivalent to the modern definition. Stated differently, explain why $$K= \lim_{A\to p}\frac{Area(n(u))}{Area(U)}=\frac{eg-f^2}{EG-F^2}$$ where e,f, and g are the coefficients of the second fundamental form and E,F,G are the coefficients of the first fundamental form. With the help of this question and my class notes, I've been able to work out a solution up to the point where the integral of a cross product comes into play. In other words, I'm not sure how the integral is evaluated to get $$\int \int_{V_{\epsilon}}||X_u \times X_v|| du dv = ||X_u(u_{\epsilon},v_{\epsilon}) \times X_v(u_{\epsilon},v_{\epsilon})||$$ and $$\int \int_{V_{\epsilon}}||N_u \times N_v|| du dv = |K(u^{'}_{\epsilon},v^{'}_{\epsilon})|||X_u(u^{'}_{\epsilon},v^{'}_{\epsilon}) \times X_v(u^{'}_{\epsilon},v^{'}_{\epsilon})||$$
I looked at both this link and this one in hopes of figuring it out on my own. Based on those I was hoping the derivative of $||X\times X_v||$ would be the result of the first integral, but the derivative gives $||X_u\times X_v||+||X\times X_{uv}||$ and I cannot assume that $X_{uv}$ is zero.