Suppose we have a surface parametrized by $\sigma$, (unknown) but suppose we also have parametrized surface $\sigma_0$ known I'm aiming to minimizing an integral involving a term like
$$ F_I = \begin{pmatrix} \lVert \sigma_u \rVert^2 & \left\langle \sigma_u, \sigma_v \right\rangle \\ \left\langle \sigma_u, \sigma_v \right\rangle &\lVert \sigma_v \rVert^2 \end{pmatrix} $$
Like wise we have $F_I^0$ for $\sigma_0$. I want to preserve local isometries between $\sigma$ and $\sigma_0$ and the only way I thought to achieve this was to use an integral
$$ E(F_I) =\int_U \lVert F_I - F_I^0 \rVert^2_2 dU = \int_U \mathcal{L}(u,v,\sigma_u,\sigma_v)dU $$
Which makes sense to me, I would be able to differentiate the integral (i.e. Euler Lagrange equations) If I were using the Frobenious norm, but I'm assuming norm $2$, i.e. max eigenvalues of $A^T A$, $A$ 2x2 matrix.
The question is, is there an easy way to differentiate that integral? namely how do I compute both
$$ \frac{d}{du} \frac{\partial}{\partial \sigma_u} \mathcal{L} = ? \\ $$
and
$$ \frac{d}{dv} \frac{\partial}{\partial \sigma_v} \mathcal{L} = ? $$
I need a suggestion/guideline/insight to solve this.