Editing Second fundamental form (section)

== Surface in R<sup>3</sup> ==
[[File:Second fundamental form.svg|300px|thumb|right|Definition of second fundamental form]]

===Motivation===
The second fundamental form of a [[parametric surface]] {{math|''S''}} in {{math|'''R'''<sup>3</sup>}} was introduced and studied by [[Carl Friedrich Gauss|Gauss]]. First suppose that the surface is the graph of a twice [[continuously differentiable]] function, {{math|''z'' {{=}} ''f''(''x'',''y'')}}, and that the plane {{math|''z'' {{=}} 0}} is [[tangent]] to the surface at the origin. Then {{math|''f''}} and its [[partial derivative]]s with respect to {{math|''x''}} and {{math|''y''}} vanish at (0,0). Therefore, the [[Taylor expansion]] of ''f'' at (0,0) starts with quadratic terms:

: <math> z=L\frac{x^2}{2} + Mxy + N\frac{y^2}{2} + \text{higher order terms}\,,</math>

and the second fundamental form at the origin in the coordinates {{math|(''x'',''y'')}} is the [[quadratic form]]

: <math> L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,. </math>

For a smooth point {{math|''P''}} on {{math|''S''}}, one can choose the coordinate system so that the plane {{math|''z'' {{=}} 0}} is tangent to {{math|''S''}} at {{math|''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 {{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>

===Physicist's notation===
The second fundamental form of a general parametric surface {{math|''S''}} is defined as follows.

Let {{math|'''r''' {{=}} '''r'''(''u''<sup>1</sup>,''u''<sup>2</sup>)}} 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''<sup>''α''</sup>}} by {{math|'''r'''<sub>''α''</sub>}}, {{math|α {{=}} 1, 2}}. Regularity of the parametrization means that {{math|'''r'''<sub>1</sub>}} and {{math|'''r'''<sub>2</sub>}} are linearly independent for any {{math|(''u''<sup>1</sup>,''u''<sup>2</sup>)}} 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>1</sub> × '''r'''<sub>2</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}_1\times\mathbf{r}_2}{|\mathbf{r}_1\times\mathbf{r}_2|}\,.</math>

The second fundamental form is usually written as

:<math>\mathrm{I\!I} = b_{\alpha \beta} \, du^{\alpha} \, du^{\beta} \,.</math>

The equation above uses the [[Einstein notation|Einstein summation convention]].
 
The coefficients {{math|''b''<sub>''αβ''</sub>}} at a given point in the parametric {{math|''u''<sup>1</sup>''u''<sup>2</sup>}}-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 in terms of the normal vector {{math|'''n'''}} as follows:

:<math>b_{\alpha \beta} = r_{,\alpha \beta}^{\ \ \,\gamma} n_{\gamma}\,. </math>