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 Last modified: 03:49 PM - 13 November, 2022

1. Consider any surface \(S\subset \mathbb{R}^3\), let \(f : S \to \mathbb{R}\), \(f(x,y,z) = x\). Show that this is a smooth function.

2. Consider a surface \(S\subset \mathbb{R}^3\), let \(f : S \to P\), where \(f(x,y,z) = (x,y,0)\) and \(P\) is the plane defined by \(z=0\). Show that this is a smooth function.

3. Give a surface patch for a sphere. Compute the first fundamental form using the surface patch.

4. Give a surface patch for a plane. Compute the first fundamental form using the surface patch.

5. How does the matrix associated with the first fundamental form change under a coordinate transformation?

6. For the map \(f\) in exercise 2, and compute \(\mathcal{D}_p(f)\).

7. This exercise is to revise what is taught in the lecture. Recall the definition of the Weingarten map, which is denoted by \(\mathcal{W}_p\)

   1. Prove that \(\mathcal{W}_p(\sigma_x)=\hat{\tilde{\mathbf{n}}}_x\) and \(\mathcal{W}_p(\sigma_y)=\hat{\tilde{\mathbf{n}}}_y\)

   2. Prove that \(\mathcal{W}(\mathbf{v})\) lies in the tangent space for any tangent vector \(v\).

   3. Prove that the Weingarten map is a linear map.

   4. Therefore, it is enough to compute the Weingarten map for the basis tangent vectors \(\sigma_x\) and \(\sigma_y\) and we therefore need to find the coefficients \(a, b, c\) and \(d\), below: \[\mathcal{W}_p(\sigma_x)=a\sigma_x+b\sigma_y\] \[\mathcal{W}_p(\sigma_y)=c\sigma_x+d\sigma_y\]

   5. Define \(L:=\sigma_{xx} . \hat{\mathbf{n}}\), \(M:=\sigma_{xy} . \hat{\mathbf{n}}\), \(N:=\sigma_{yy} . \hat{\mathbf{n}}\). Prove that \(L= - \sigma_x.\hat{\mathbf{n}}_x\), \(M= - \sigma_x.\hat{\mathbf{n}}_y = - \sigma_y.\hat{\mathbf{n}}_x\), and \(N= - \sigma_y.\hat{\mathbf{n}}_y\). The following matrix is called the matrix of the matrix of the second fundamental form: \[\begin{pmatrix} L & M\\ M & N \end{pmatrix}\]

   6. By taking the dot product on both sides of each of the two equations in part 4 by \(\sigma_x\) and then by \(\sigma_y\), obtain 4 linear equations whose unknowns are \(a\), \(b\), \(c\), and \(d\). Compute \(a\), \(b\), \(c\), and \(d\) in terms of the first fundamenal form (i.e. \(E\), \(F\), \(G\)), and the second fundamental form (\(L\), \(M\), and \(N\)).