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 Last modified: 03:04 AM - 20 November, 2021

For all these questions, \(\mathbf{\hat{n}}(p):=\sigma_x(p)\times \sigma_y(p)\)

1. Prove that \(\mathbf{\hat{n}}(\gamma(t))\) is perpendicular to \(\mathbf{T}(t)\).

2. Prove that \(\mathbf{N}(t) = \mathbf{\hat{n}}(t)\) if and only if \(\kappa_g(t)=0\)

3. Prove that the area can be expressed entirely in terms of the first fundamental form. \[A_\sigma(R) = \int_R \sqrt{E(x,y) G(x,y) - F^2(x,y)}\mathrm{d}x\mathrm{d}y\]

4. How does the matrix of the first fundamental form vary with a coordinate transformation?

5. Prove that if the surface patch is regular then the matrix of the first fundamental form is invertible.

6. Recall the definition of \(D_p(f) : T_p(S) \to T_p(S)\)

   1. Show that \(D_p(Id_S) = Id_{T_p(S)}\)

   2. Show that \(D_p(f \circ g) = D_p(f) \circ D_p(g)\) where \(g : S_1 \to S_2\) and \(f : S_2 \to S_3\) are smooth functions between surfaces.

   3. Prove that if \(f\) is smooth with a smooth inverse, then \(D_p(f)\) is invertible.

7. Prove that \(\mathcal{W}\mathbf{v}.\mathbf{w} = \mathcal{W}\mathbf{w}.\mathbf{v}\)