Editing Second fundamental form (section)

===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>