Homework 12

Due Date: Friday, December 12

1. Find the endpoints of the major and minor axes of the ellipse \(8x^2 + 12xy + 17y^2 = 100\).
2. Let \(f(x,y) = 3x^2 + 4\cos(x+y)\), and let \(S\) be the surface \(z = f(x,y)\), with normal vectors pointing upwards.
   1. Show that \((0,0)\) is a critical point for \(f\), and use the second derivative test to determine whether \((0,0)\) is a local minimum, a local maximum, or a saddle point.
   2. Compute the principle curvatures, Gaussian curvature, and mean curvature for \(S\) at the point \((0,0,4)\).
3. Let \(S\) be the paraboloid \(z = x^2 + y^2\), oriented with normal vectors pointing upwards, and consider the curve \(\vec{x}(t) \;=\; (1,t,1+t^2)\).
   1. Compute the curvature \(\kappa\) and principle normal vector \(\vec{P}\) for this curve at the point \((1,0,1)\).
   2. Compute the Darboux frame \(\{\vec{N},\vec{T},\vec{U}\}\) for this curve at the point \((1,0,1)\).
   3. Compute the normal curvature \(\kappa_n\) and geodesic curvature \(\kappa_g\) for this curve at the point \((1,0,1)\)
4. Let \(S\) be the torus \(\vec{X}(u,v) = \bigl((2+\cos u)\cos v,(2+\cos u)\sin v,\sin u\bigr)\), oriented with normal vectors pointing outwards, and let \(p\) be the point \((2+\cos u,0,\sin u)\).
   1. Compute the normal vector \(\vec{N}\) to this torus at the point \(p\).
   2. Compute \(\vec{T}\), \(\vec{U}\), \(\kappa\), \(\vec{P}\), \(\kappa_n\), and \(\kappa_g\) for the \(u\)-coordinate line at the point \(p\).
   3. Compute \(\vec{T}\), \(\vec{U}\), \(\kappa\), \(\vec{P}\), \(\kappa_n\), and \(\kappa_g\) for the \(v\)-coordinate line at the point \(p\).
   4. Compute the principle curvatures, Gaussian curvature, and mean curvature for \(S\) at the point \(p\).
   5. For what values of \(u\) is \(K>0\)? For what values of \(u\) is \(K<0\)?
   6. Compute \(\displaystyle\int\!\!\!\int_S K\,dA\) and \(\displaystyle\int\!\!\!\int_S |K|\,dA\).