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\leftline{MATH 392 -- Geometry Through History -- Makeup Quiz 2 -- April 18, 2016}
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\leftline{Your Name: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$}
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\noindent
Let $S$ be the surface parametrized by
$$\vec{x}(u,v) = (2u,\sinh(u)\cos(v),\sinh(u)\sin(v)),\quad (u,v) \in (0,\infty) \times (-\infty,\infty)$$
where the function $\sinh(u) = \frac{e^u - e^{-u}}{2}$
is the hypberbolic sine.  (The identity
$$\cosh^2(u) - \sinh^2(u) = 1$$
may be useful!)  
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\item[(A)] (5)  Compute the coefficients $E,F,G$ of the first fundamental form.
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\item[(B)] (10) Compute the coefficients $e,f,g$ of the second fundamental form.
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\item[(C)] (5) Find the Gaussian curvature of $S$ (as a function of $u,v$).  
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