Editing Second fundamental form (section)

===Classical notation===
The second fundamental form of a general parametric surface is defined as follows. Let {{math|1='''r''' = '''r'''(''u'',''v'')}} be a regular parametrization of a surface in {{math|'''R'''<sup>3</sup>}}, where {{math|'''r'''}} is a smooth [[vector-valued function]] of two variables. It is common to denote the partial derivatives of {{math|'''r'''}} with respect to {{math|''u''}} and {{math|''v''}} by {{math|'''r'''<sub>''u''</sub>}} and {{math|'''r'''<sub>''v''</sub>}}. Regularity of the parametrization means that {{math|'''r'''<sub>''u''</sub>}} and {{math|'''r'''<sub>''v''</sub>}} are linearly independent for any {{math|(''u'',''v'')}} in the domain of {{math|'''r'''}}, and hence span the tangent plane to {{math|''S''}} at each point. Equivalently, the [[cross product]] {{math|'''r'''<sub>''u''</sub> × '''r'''<sub>''v''</sub>}} is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors {{math|'''n'''}}:

:<math>\mathbf{n} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|} \,.</math>

The second fundamental form is usually written as

:<math>\mathrm{I\!I} = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,,</math>

its matrix in the basis {{math|{'''r'''<sub>''u''</sub>, '''r'''<sub>''v''</sub><nowiki>}</nowiki>}} of the tangent plane is

:<math> \begin{bmatrix}
L&M\\
M&N
\end{bmatrix} \,. </math>

The coefficients {{math|''L'', ''M'', ''N''}} at a given point in the parametric {{math|''uv''}}-plane are given by the projections of the second partial derivatives of {{math|'''r'''}} at that point onto the normal line to {{math|''S''}} and can be computed with the aid of the [[dot product]] as follows:

:<math>L = \mathbf{r}_{uu} \cdot \mathbf{n}\,, \quad
M = \mathbf{r}_{uv} \cdot \mathbf{n}\,, \quad
N = \mathbf{r}_{vv} \cdot \mathbf{n}\,. </math>

For a [[Signed distance function|signed distance field]] of [[Hessian matrix|Hessian]] {{math|'''H'''}}, the second fundamental form coefficients can be computed as follows:

:<math>L = -\mathbf{r}_u \cdot \mathbf{H} \cdot \mathbf{r}_u\,, \quad
M = -\mathbf{r}_u \cdot \mathbf{H} \cdot \mathbf{r}_v\,, \quad
N = -\mathbf{r}_v \cdot \mathbf{H} \cdot \mathbf{r}_v\,. </math>