Homework 11

Due Date: Friday, December 5

1. Let \(S_1\) be the cylinder \(\vec{X}(u,v) = (\cos u,\sin u,v)\) for \( 0 < u < 2\pi \) and \( v > 0 \), let \(S_2\) be the paraboloid \(z=x^2+y^2\), and let \(f\colon S_1\to S_2\) be the map \(f(x,y,z) = (xz,yz,z^2)\).
   1. Compute \(\vec{X}_u\) and \(\vec{X}_v\) at the point \(p = (0,1,3)\).
   2. Compute \(f(p)\), \(df_p\bigl(\vec{X}_u\bigr)\), and \(df_p\bigl(\vec{X}_v\bigr)\), where \(p=(0,1,3)\).
   3. Compute \(df_p(1,0,1)\), where \(p=(0,1,3)\).
2. Let \(S\) be the cone \(z=\sqrt{x^2+y^2}\) for \(z>0\). Classify each of the following maps as equiareal, conformal, both, or neither.
   1. The map \(f\colon S\to S\) defined by \(f(x,y,z) = (3x,3y,3z)\).
   2. The map \(f\colon S\to S\) defined by \(f(x,y,z) = (-y,x,z)\).
   3. The map \(f\colon S\to S\) defined by \(f(x,y,z) = (xz,yz,z^2)\).
   4. The map \(f\colon S\to S\) defined by \(\displaystyle f(x,y,z) = \frac{\sqrt{1+z^2}}{z}(x,y,z)\).
3. Let \(S_1\) be the sphere \(x^2+y^2+z^2 = 4\), let \(S_2\) be the ellipsoid \((x/6)^2 + (y/8)^2 + (z/10)^2 = 1\), and let \(f\colon S_1\to S_2\) be the map \(f(x,y,z) = (3x,4y,5z)\). Compute the Jacobian of \(f\) at the point \((2,0,0)\).
4. Let \(S\) be the catenoid \(r = \cosh z\), oriented so that the normal vectors point outwards (away from the \(z\)-axis).
   1. Compute the function \(\vec{N}(u,v) = n\bigl(\vec{X}(u,v)\bigr)\), where \(\vec{X}\) is the parametrization \(\vec{X}(u,v) = (\cosh u \cos v,\cosh u \sin v, u)\).
   2. Compute the absolute Gaussian curvature \(|K(u,v)|\).
   3. Use your formula from part (b) to evaluate the integral \(\displaystyle\int\!\!\!\!\int_S |K|\, dA\).
   4. What is the image of \(S\) under the Gauss map? Does this agree with your answer to part (c)?