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The picture below shows the surface \(S\) defined as the graph of the function \(f : \mathbb{R}^2 \rightarrow \mathbb{R}^3\) given by \(f(x,y) = x^2 + ry^2\). In Lecture 3 of Week 8 we parametrised this surface by \[ p : \mathbb{R}^2 \rightarrow \mathbb{R}^3, \quad p(u,v) = \left( u, v, f(u,v) \right) . \] At the origin \(p(0,0) \in S\), the tangent space \(T_{p(0,0)} S\) has a basis \( \{p_u(0,0), p_v(0,0)\}\). With respect to this basis, we computed the shape operator as \[ A_{p(0,0)} = \left( \begin{matrix} 2 & 0 \\ 0 & 2r \end{matrix} \right) \] The eigenvalues of this matrix are \(2\) and \(2r\). When \(r > 0\) then the eigenvalues have the same sign, and so the surface has a bowl-shape at the origin (you can see this in the diagram below by choosing a positive value of \(r\)). When \(r < 0\) the eigenvalues have opposite signs, and so the surface has a saddle-shape at the origin (again, you can see this in the picture below by choosing a negative value of \(r\)).
