Homework 10

Due Date: Wednesday, November 26

1. Let \(S\) be the hyperbolic paraboloid \(z=xy\), and let \(\vec{X}\) be the parametrization \[ \vec{X}(u,v) = (u,v,uv). \]
   1. Find the first fundamental form \(g(u,v)\) of \(S\) with respect to \(\vec{X}\).
   2. Verify that \(\sqrt{\det(g)} = \|\vec{X}_u \times \vec{X}_v\|\).
2. Let \(\vec{X}\) be a surface parametrization, and suppose that the corresponding first fundamental form is \[ g(u,v) \;=\; \begin{bmatrix}1+u^2 & uv^2 \\ uv^2 & 1+v^4\end{bmatrix} \]
   1. Let \(\vec{x}(t) = (t,t^2)\), and let \(\vec{y}(t) = \vec{X}\bigl(\vec{x}(t)\bigr)\). Compute \(\|\vec{y}\,'(1)\|\).
   2. Find the angle between the curve \(\vec{y}\) and the \(u\) coordinate line at the point \(\vec{X}(1,1)\).
3. The hyperbolic plane \(\mathbb{H}^2\) is the setting for non-Euclidean geometry. It can be parameterized by a function \(\vec{X}\colon U \to \mathbb{H}^2\), where \(U\) is region \(v>0\) in the \(uv\)-plane, whose first fundamental form is \[ g(u,v) \;=\; \begin{bmatrix}1/v^2 & 0 \\ 0 & 1/v^2\end{bmatrix}. \]
   1. Let \(C_1\) be the straight-line path in the \(uv\)-plane from \((1,1)\) to \((-1,1)\). Find the the length of the corresponding curve in \(\mathbb{H}^2\).
   2. Let \(C_2\) be the arc of the circle \(u^2 + v^2 = 2\) from the point \((1,1)\) to \((-1,1)\). Find the length of the corresponding curve in \(\mathbb{H}^2\). (Feel free to use Wolfram Alpha for the integral.)
   3. Which of the curves \(C_1\) and \(C_2\) is longer on the \(uv\)-plane? Which of the corresponding curves is longer on the hyperbolic plane?
   4. Let \(R\) be the region \( 0 < u < 1 \) and \( 1 < v < 2 \) in the \( uv \)-plane. Find the area of the corresponding region in the hyperbolic plane.
   5. Is the parametrization \(\vec{X}\) equiareal? Is it conformal? Is it isometric?