Editing Second fundamental form (section)

===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.