Tangent plane

The tangent plane of a shape $S$ at $x_0 = \xi(u_0, v_0)$, denoted $T_{x_0}S$, has equation

$$T_{x_0}S = \xi(u_0, v_0) + (u - u_0) \frac{\delta\xi}{\delta u}(u_0, v_0) + (v - v_0) \frac{\delta\xi}{\delta v}(u_0, v_0)$$

The set of tangent vectors $\overrightarrow{T_{x_0}S}$ is a sub-space of dimension two of $\vec{E}$. To place a tangent vector on the shape, it must be displaced by $\xi(u_0, v_0)$.