Discussion
Let the first and second fundamental form of a surface patch are \(E\,du^{2} + 2F\,du\,dv + G\,dv^{2}\) and \(L\,du^{2} + 2M\,du\,dv + N\,dv^{2}\) respectively. Then the Gaussian curvature of the patch is :
Answer: D
The Gaussian curvature is \(\det\begin{pmatrix} L & M \\ M & N \end{pmatrix}\) / \(\det\begin{pmatrix} E & F \\ F & G \end{pmatrix}\).
Gaussian curvature is the determinant of the shape operator, which is the second fundamental form expressed relative to the first: its matrix is \(\mathcal{F}_{I}^{-1}\mathcal{F}_{II}\). Taking determinants,
\[ K = \frac{LN - M^{2}}{EG - F^{2}} \]
The determinant of the first form alone is the squared area element, and that of the second alone depends on the parametrisation; their product has no geometric meaning. Only the ratio is independent of the parametrisation.
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