Exam 2

There are three questions. Each is worth 10 points.

Please email me your solutions by the midnight of Wednesday April 3rd.

Q1

The animation below shows the surface $M$ of the graph of $z = x^2 - y^2$ plotted in the $xyz$ Cartesian coordinates. Normal sections of $M$ at the origin at varying orientations are plotted as yellow curves.

  1. Find the equation of the yellow curve as a function of the section's orientation.
  2. Calculate the curve's curvature at the origin.
  3. Plot the graph of the curvature over a range of π radians.
[normal-curvature.gif]

Q2

A circle of radius $b$ is rotated about a line, within the circle's plane, set at a distance $a$ from the circle's center, as shown in the diagram below.
[generate-torus.png]
  1. Show that $\mathbf{x}(u,v) = \begin{bmatrix} (a+b \cos v)\,\cos u \\ (a+b \cos v)\,\sin u \\ b \sin v \end{bmatrix}$ is a parametric representations of the resulting torus.
  2. Find the torus's unit normal $\mathbf{N}(u,v)$.
  3. Let $\mathcal{S}$ be the shape operator at any point on the torus. Calculate $\mathcal{S}(\mathbf{x}_u)$ and $\mathcal{S}(\mathbf{x}_v)$ and conclude that \[ \mathcal{S}(\mathbf{x}_u) = - \frac{\cos v}{a+b\cos v} \mathbf{x}_u, \quad \mathcal{S}(\mathbf{x}_v) = - \frac{1}{b} \mathbf{x}_v. \]
  4. What does this say about the eigenvalues and eigenvectors of $\mathcal{S}$?
  5. Verify that the eigenvectors of $\mathcal{S}$ are orthogonal as they should be.

Q3

Solve Exercise 2.2.16 (page 85) of the textbook.