Linear approximation of membrane energy From a reference I have in mesh processing the following well known equation is used to compute the area of a parametric surface
$$
A(\sigma) = E_M(\sigma) = \int_U \sqrt{EF - G^2}dU
$$
Which is also called membrane energy, $E,F$ and $G$ are the coefficients of the first fundamental form. However minimizing such form, even numerically, is difficult, therefore usually the following linearization is proposed
$$
\tilde{E}_M(\sigma) = \int_U \lVert \sigma_u \rVert^2 + \lVert \sigma_v\rVert^2 dU
$$
However I do struggle to understand why this linearization is true, can anyone suggest how to get the second integral from the first?
 A: Disclaimer: I have no idea what people do in numerics to find minimal surfaces, however, I can give an answer to the analytical side of your question.
There have been quite a few different methods for solving the Plateau problem. 
In general, we have the inequality
$$\begin{align*}\|\sigma_u\wedge \sigma_v\|
=\sqrt{\|\sigma_u\|^2\|\sigma_v\|^2-\langle\sigma_u,\sigma_v\rangle^2}
&=\sqrt{EF-G^2} \\
&\leq \frac12(\|\sigma_u\|^2+\|\sigma_v\|^2)
=\frac12\|(\sigma_u,\sigma_v)\|^2
=\frac12\|\nabla\sigma\|^2
\end{align*}
$$
with equality if and only if $\|\sigma_u\|=\|\sigma_v\|$ and $\langle\sigma_u,\sigma_v\rangle=0$. 
(You can see this by squaring and rearranging the inequality, to get the equivalent inequality which trivially holds true,
$$
0
\leq
\langle\sigma_u,\sigma_v\rangle^2 + (\|\sigma_u\|^2 - \|\sigma_v\|^2)^2
$$
with equality as above.)
A parametric surface $\sigma\colon B\subset\mathbb{R}^2\to \mathbb{R}^n$ which fulfils the two conditions above is called conformally parametrised.
Thus, in general, the area functional $A(\sigma)=\int_B \|\sigma_u\wedge \sigma_v\| dudv$ is dominated by the so-called Dirichlet energy $D(\sigma)=\frac12 \int_B \|\nabla\sigma\|^2dudv$ with equality if and only if $\sigma$ is conformally parametrised.
Classically, you would first try to minimise the Dirichlet energy because the energy integrand has better regularity properties. For instance, it is continuous on all of $\mathbb{R}^{n\times 2}$ and $C^1$ on $\mathbb{R}^{n\times 2}\setminus\{0\}$. Another important feature of $D$ is the invariance under conformal reparametrisations of the domain $B$.
This contrasts the area functional, whose integrand is only $C^1$ on the smaller set of rank-2 matrices but which is invariant under the larger class of diffeomorphic reparametrisations (which is a bad thing).
You can use a compactness argument to find a minimiser $\sigma$ of $D$. Subsequently, when you do an inner variation of the energy functional you realise that vanishing inner variation (due to the minimality) of the energy already means that the surface $\sigma$ is conformally parametrised!
Hence, you know that area functional and the Dirichlet energy of $\sigma$ yield the same value and so you have found a surface of minimal energy which also is an area minimiser.
For instance, this is how they do it in Chapter 4 of the book Minimal surfaces I by Dierkes, Hildebrandt, Küster and Wohlrab.
A: It's not a ‘true’ linearization. Moreover, one can show (from the dimension analysis) that you can scale one of the parameters $u$ or $v$ to make $\tilde E_M$ as bad as you want.
However, if we believe that our maps are square enough (i.e. $\lVert\sigma_u\rVert\approx\lVert\sigma_v\rVert$ and $(\sigma_u,\sigma_v)\approx0$), then:
$$
\sqrt{EG-F^2} \approx \lVert\sigma_u\rVert\cdot\lVert\sigma_v\rVert\approx\frac12\left(\lVert\sigma_u\rVert^2 + \lVert\sigma_v\rVert^2\right)
$$
