Reading: Syllabus. Text: Chapter 5 through the Fundamental Theorem.
Problems: Chapter 5: 1-4, 6 [Note: Read Problem 6 as follows: Take any circle of radius R=a with center on the y-axis. Show that the lower half of the circle meets the tractrix at a right angle.]
Problem A: Let α(t) be a curve in Rn parametrized by arc length and F(x) = Ax + b a Euclidean motion (i.e., A is an orthogonal n×n matrix, which means A AT = I, and b is a vector). Show that the curve F ◦α is also parametrized by arc length. [Here we think of all vectors as column-vectors.]
Problem B: Under the assumptions of Problem A, when n=2, i.e., the curve α is a plane curve, show that the curves α and F ◦ α have the same curvature.
Problem C: Let α(t) be a unit-speed plane curve. Suppose it lies inside the disk of radius R around the origin, that is to say, ||α(t)||≤ R. Suppose that α(t) touches the boundary of the disk at t=t0, that is to say, ||α(t0)|| = R. Show that the curvature κ of α at this point satisfies the inequality: |κ (t0)| ≥ 1/R. [Hint: Study ||α(t)||2 as a function of t. Compute the first two derivatives of this function at t = t0 and apply the second-derivative test.]
Last modified: (2013-02-16 03:13:20 CST)