Tangent deformation

Let $\vec{\tau}_{t_i}$ be a tangent vector of the shape $S_{t_i}$:

$$\vec{\tau}_{t_i}(u_0, v_0) = du \frac{\delta\xi}{\delta u}(u_0, v_0) + dv \frac{\delta\xi}{\delta v}(u_0, v_0)$$

This vector is transformed into $\vec{\tau}_{t_{i+1}}$ of the shape $S_{t_{i+1}}$:

$$\begin{array}{rcl} \vec{\tau}_{i+1}(u_0, v_0) &=& du \frac{\d... ... t_{i+1}}}{\delta \xi}(x_0) \vec{\tau}_i(u_0, v_0) \end{array}$$

Where $J = \frac{\delta f_{t_i\mapsto t_{i+1}}}{\delta \xi}(x_0)$ is the Jacobian of $f_{t_i\mapsto t_{i+1}}$. Thus tangent vectors are transformed by the Jacobian.