I need general indications or guidance.
I do not know how to map a surface $ z = \sin (\pi x) \sin (\pi y), (x,0,\pi), (y,0,\pi)$ to a unit square. Nor do I know how to map a quadrant of a unit sphere:
$$ (x_1,y_1,z_1) = ( \cos(u) \cos(v), \cos(u)\sin(v), \sin(u)),\quad ( u,0, \pi/2), (v,0,\pi/2)$$ with metric $ ds^2 = du^2 + \cos^2(u) dv^2 $ onto a square flat patch $$ (x_2,y_2,z_2) = ( u \pi/2 ,v \pi/2, 0 ), \quad (u,0,1), (v,0,1), $$ where one edge totally shrinks to zero in length.
Since it does not preserve lengths / angles, there is no isometry but E,F,G etc. need to be further differentiated and set to future strained shape. I seek to find what relations exist between coefficients of first/second fundamental forms and their successive partial derivatives that bring about strain/dilation, shown by change of length and area differentials?
I have in mind Beltrami's first differential parameter, i.e., Laplacian of Gauss curvature and also the von Karman's strain compatibility relations from Laminate theory in mechanics of materials.
At least, let me know where my question makes no sense.