MATH 396 PROBLEMS 5

IGOR KRIZ

Regular problems:

1. (a) Find a way to represent the unit circle {(x, y)|x^{2} + y^{2} = 1} as a piecewise smooth curve.

(b) Calculate the curve integral of the first kind (=line integral with respect to length) of the function f(x, y) = |xy| over the unit circle.

2. Find a piecewise smooth curve whose set of points is the intersection of the two surfaces {(x, y, z)|x^{2} + y^{2} = z^{2}} and {(x, y, z)|y^{2} = z}.

3. Calculate the curve integral of the first kind (=line integral with respect to length) of the function f(x, y) = x over the segment of a parabola {(x, x^{2})|0 ≤ x ≤ 1}.

4. Calculate the curve integral of the first kind (=line integral with respect to length) of the function f(x, y) = x + y over the triangle in the plane with vertices (0, 0), (1, 0) and (0, 1).

Challenge problems:

5. Prove in (more) detail that there exists a Peano curve, i.e. a continuous function f : [0, 1] → ℝ^{2} completely filling a square. [Construct - or show the existence of - a sequence of functions uniformly converging to the curve; use compactness.]

6. Two pieces of rope (wick), when lit on one side, take exactly 1 hour each to burn across. However, the speed of burning is not constant, and the ropes are not identical. Can you find a way to measure exactly 45 minutes using the ropes?