Intuition behind the arc length of a curve on the surface and the area of a surface expressed by using first fundamental form I want to understand the following formulas for defining the arc length of a curve on the surface and the area of a surface from wikipedia:

(1)$ds^2 = Edu^2+2Fdudv+Gdv^2,$
(2)$dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \langle X_u,X_v \rangle^2 } \ du\, dv = \sqrt{EG-F^2} \, du\, dv.$

I do understand the following formulas for the arc length and area(proved by chopping the arc or surface into small line segments or flakes):

(3)$L = \int_a^b \sqrt{ (x'(t))^2+(y'(t))^2 + (z'(t))^2 } dt$
(4)$A=\iint_D \sqrt{ 1 + (f'(x))^2+(f'(y)) } dxdy$


Updated at 11:24pm:
Somehow I can understand (1) from (3) now.
But the understanding of (4) doesn't help me understand (2). What I want is a physicists' "proof" of (2), which may not be rigorous, but well explains the intuition behind the formula (2).
Any solution or reference will be appreciated!
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$On a surface, a coordinate system $(u, v)$ gives rise to coordinate vector fields $\dd_{u}$ and $\dd_{v}$ and coordinate differentials $du$ and $dv$. The metric $g$ has components
\begin{align*}
g_{11} = E &= g(\dd_{u}, \dd_{u}) = \|\dd_{u}\|^{2}, \\
g_{22} = G &= g(\dd_{v}, \dd_{v}) = \|\dd_{v}\|^{2}, \\
g_{12} = F &= g(\dd_{u}, \dd_{v}) = \|\dd_{u}\|\, \|\dd_{v}\| \cos\theta,
\end{align*}
with $\theta$ denoting the intrinsic angle between the $u$- and $v$-coordinate curves.
The area of an infinitesimal parallelogram with sides $\dd_{u}$ and $\dd_{v}$ is
$$
\|\dd_{u}\|\, \|\dd_{v}\| \sin\theta
  = \sqrt{\|\dd_{u}\|^{2}\, \|\dd_{v}\|^{2} (1 - \cos^{2}\theta)}
  = \sqrt{EG - F^{2}},
\tag{1}
$$
so that the $2$-form
$$
dA = \sqrt{EG - F^{2}}\, du\, dv
$$
measures the area of an arbitrary infinitesimal parallelogram spanned by an ordered pair of tangent vectors.
If the surface is immersed in Euclidean $3$-space by a mapping $X$ defined in some plane region, then $\dd_{u} = X_{u}$ and $\dd_{v} = X_{v}$ may be viewed as vectors in $\Reals^{3}$. The area of the parallelogram they span is the magnitude of their cross product:
$$
\|X_{u} \times X_{v}\| = \|X_{u}\|\, \|X_{v}\| \sin\theta,
$$
compare (1).

