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The helicoid can be described by \[ {\Phi }(u, v) = (u \cos v, u \sin v, bv), \hbox{where } b \neq 0. \]
Show that \(H = 0\) and that \(K = - b^{2}/(b^{2}+u^{2})^{2}\). In Figure 17.52 and Figure 17.56, we see that the helicoid is actually a soap film surface. Surfaces in which \(H = 0\) are called minimal surfaces.
Consider the saddle surface \(z\,{=}\,{\it xy}\). Show that \[ K= \frac{-1}{(1 + x^2 +y^2)^2}, \] and that \[ H= \frac{-{\it xy}}{(1+ x^2 +y^2)^{3/2}}. \]
Show that \({\Phi} (u, v) = (u, v, \log \cos v - \log \cos u)\) has mean curvature zero (and is thus a minimal surface; see Exercise 1).
Find the Gauss curvature of the elliptic paraboloid \[ z= \frac{x^2}{a^2} + \frac{y^2}{b^2}. \]
Find the Gauss curvature of the hyperbolic paraboloid \[ z= \frac{x^2}{a^2} - \frac{y^2}{b^2}. \]
Compute the Gauss curvature of the ellipsoid \[ \frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2}=1. \]
After finding \(K\) in the previous exercise, integrate \(K\) to show that: \[ \frac{1}{2 \pi} {\int\!\!\!\int}_{S} K \,dA = 2. \]
Find the curvature \(K\) of:
Show that Enneper’s surface \[ {\Phi}(u, v) = \left(u-\frac{u^3}{3} + uv^2, v - \frac{v^3}{3} + u^2 v, u^2 - v^2\right) \] is a minimal surface (\(H = 0\)).
Consider the torus \(T\) given in the fourth exercise, Section 17.4. Compute its Gauss curvature and verify the theorem of Gauss–Bonnet. [HINT: Show that \(\| T_{\theta } \times T_{\phi }\| ^{2} = (R + \cos \phi)^{2}\) and \(K = \cos \phi/(R + \cos \phi)\).]
Let \({\Phi}(u, v) = (u, h(u) \cos v, h(u) \sin v), h \,{>}\, 0\), be a surface of revolution. Show that \(K = - h^{\prime\prime}/h {\{}1 + (h^{\prime} )^{2}{\}}^{2}\).
A parametrization \({\Phi}\) of a surface \(S\) is said to be conformal (see Section 17.4), provided that \(E=G\), \(F\) = 0. Assume that \({\Phi}\) conformally parametrizes \(S\).footnote # Show that if \(H\) and \(K\) vanish identically, then \(S\) must be part of a plane in \({\mathbb{R}}^{3}\).