Surfaces deformation

Let $f = f_{t_i\mapsto t_{i+1}}$ be a transformation of the shape. First, note that the transformations of the first and second derivatives of a parameterized surface are given by the Jacobian $J = (\frac{\delta f}{\delta x}, \frac{\delta f}{\delta y}, \frac{\delta f }{\delta z})$:

$$\begin{array}{rcl} \frac{\delta f\circ\xi}{\delta u} &=&\fra... ...{\delta\xi} \frac{\delta^2\xi}{\delta u\delta v}\\ \end{array}$$