Reading: Chapter 6 through page 109.
Problems: Chapter 6: 1 [Establish Property (4) from page 99 only and do not do the Lagrange formula in the exercise], 2 [Just do Statement (3): lines can be characterized as space curves of zero curvature], 3, 4, 12.
Problem D: The curve formed by centers of osculating circles of a curve α is called the evolute of α. Show that the evolute of the parabola y = x2 is given by the equation (y-1/2)3 = 27/16 x2.
Problem E: Show that the torsion of a space curve is invariant under an orientation preserving Euclidean motion F(x) = Ax + b with A being a special orthogonal 3×3 matrix, which means A At = I with det A = 1, and b being a vector.
Problem F: Let α: R → R3 be a closed, smooth, parametrized curve with period L. Let e be a unit vector. We can try to count the local maxima of α in direction e:
μ (α, e) = | {local maxima in [0,L) of the function f(t) = α(t) ⋅ e: R → R}|,
should this number happen to be finite. Here the absolute value sign means the number of elements in a set - in this case, the number of points c on [0,L) at which f(t) has a local maximum. Show that μ (α, e) = μ (α, -e), provided that all critical points c of f(t) are nondegenerate, that is to say, f"(c) ≠ 0. [Hint: Look at f'(t) and show that between any two local maxima there is always a local minimum and vice versa. How is μ (α, -e) related to the minima of f?]
Last modified: (2013-02-06 11:05:39 CST)