The shape operator $S$ of a surface $\Sigma$ in ${\mathbb{R}}^3$ is the derivative of the sphere map $N:\Sigma\to S^2$ given by $N(p)=$ unit normal vector field at $p$ So at each $p$ $S(p)=d_pN$ and it is the linear transformation $S(p):T_p\Sigma\to T_{N(p)}S^2$ This is important, because the determinant defines the Gaussian curvature at $p$ in $\Sigma$