For each of the following functions \(f\colon\mathbb{R}^3\rightarrow\mathbb{R},\) determine for which values of \(c\in\mathbb{R}\) we have that \(f^{-1}(\{c\})\) is a smoothly embedded surface. Further, when \(f^{-1}(\{c\})\) is a smoothly embedded surface, list the possible orientations.

1. \(f(x,y,z)=(x+y+z-r)^2,\) for some constant \(r\in\mathbb{R}.\)

2. \(f(x,y,z)=xyz^2.\)

3. \(f(x,y,z)=z-x^2+y^2.\)

A smoothly embedded surface \(M\subset\mathbb{R}^3\) is said to be connected if any two of its points can be joined by a continuous curve in \(M.\)

1. Show that the set defined by \(\{(x,y,z)\in\mathbb{R}^3|1=z^2-x^2-y^2\}\) is a smoothly embedded surface. Further, show that is not connected.

2. Suppose \(M \subset \mathbb{R}^3\) is a connected smoothly embedded surface. If \(f\colon M \rightarrow \mathbb{R}\) is a nonzero continuous function, then \(f\) does not change sign on \(M.\)

Show that geodesics are not globally length minimising in the following sense: There exists an embedded surface \(M\subset \mathbb{R}^3,\) a geodesic \(\gamma : [a,b] \to M,\) times \(t_0,t_1 \in [a,b]\) so that \[\int_{t_0}^{t_1}\Vert \dot{\gamma}(t)\Vert \mathrm{d}t\] is strictly larger than the length of a smooth immersed curve \(\delta : [0,1] \to M\subset \mathbb{R}^3\) with \(\delta(0)=\gamma(t_0)\) and \(\delta(1)=\gamma(t_1).\)