Discussion
Let the first and second fundamental forms associated with a surface patch be respectively \(Edu^{2} + 2F\,dudv + Gdv^{2}\) and \(Ldu^{2} + 2M\,dudv + Ndv^{2}\). If \(S = \begin{bmatrix} E & F \\ F & G \end{bmatrix}\) and \(T = \begin{bmatrix} L & M \\ M & N \end{bmatrix}\), then :
Answer: B
The answer is S is non-singular and T may not be.
The first fundamental form is \(E = \sigma_{u}\cdot\sigma_{u}\), \(F = \sigma_{u}\cdot\sigma_{v}\), \(G = \sigma_{v}\cdot\sigma_{v}\). For a regular patch \(\sigma_{u}\) and \(\sigma_{v}\) are linearly independent, so by Lagrange's identity \(EG - F^{2} = |\sigma_{u} \times \sigma_{v}|^{2} \gt 0\): S is positive definite and non-singular.
The second fundamental form can be degenerate. For a plane, \(L = M = N = 0\), and for a cylinder \(LN - M^{2} = 0\), so T is singular there. Since \(\det T = K\,\det S\), T is singular exactly where the Gaussian curvature is 0. So T may be singular, but S never is.
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