in differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional euclidean space, usually denoted by \mathrm{i\!i} (read "two"). together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. more generally, such a quadratic form is defined for a smooth hypersurface in a riemannian manifold and a smooth choice of the unit normal vector at each point. == surface in r3 == ===motivation=== the second fundamental form of a parametric surface s in r3 was introduced and studied by gauss. first suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. then f and its partial derivatives with respect to x and y vanish at (0,0). therefore, the taylor expansion of f at (0,0) starts with quadratic terms: : z=l\frac{x^2}{2} + mxy + n\frac{y^2}{2} + \mathrm{\scriptstyle{{\ }higher{\ }order{\ }terms}}, and the second fundamental form at the origin in the coordinates x, y is the quadratic form : l \, \text{d}x^2 + 2m \, \text{d}x \, \text{d}y + n \, \text{d}y^2. \, for a smooth point p on s, one can choose the coordinate system so that the coordinate z-plane is tangent to s at p and define the second fundamental form in the same way. ===classical notation=== the second fundamental form of a general parametric surface is defined as follows. let  be a regular parametrization of a surface in r3, where r is a smooth vector valued function of two variables. it is common to denote the partial derivatives of r with respect to u and v by ru and rv. regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to s at each point. equivalently, the cross product ru&nbsp;&times;&nbsp;rv is a nonzero vector normal to the surface. the parametrization thus defines a field of unit normal vectors n: :\mathbf{n} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|}. the second fundamental form is usually written as :\mathrm{i\!i} = l\, \text{d}u^2 + 2m\, \text{d}u\, \text{d}v + n\, \text{d}v^2, \, its matrix in the basis {ru, rv} of the tangent plane is : \begin{bmatrix} l&m\\ m&n \end{bmatrix}. the coefficients l, m, n at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to s and can be computed with the aid of the [[dot produc
Curvature_(mathematics) Differential_geometry Differential_geometry_of_surfaces Riemannian_geometry