CHALLENGE PROBLEMS FOR CALCULUS III STUDENTS!!!

Challenge Problem 1
Assume that P, Q, R and S are four points in space which lie in the same plane, and which determine a convex quadrilateral with diagonals vec[PR] and vec[QS]. Show that the area of the quadrilateral is (1/2)*|vec[PR] x vec[QS]|.

Challenge Problem 2
Establish the Law of Sines using cross products.

Challenge Problem 3
Assume that p is a point on curve C_1 and on curve C_2. Assume that both curves have the same TNB-frame at point p, and that the curvature and torsion of the two curves coincide for all values of s (s = arc length parameter measured from point p). Show that C_1 = C_2. You may assume as much smoothness as you need to perform any differentiation required.

Hints:
• Let < T,N,B > and < t,n,b > be the respective TNB-frames for the two curves. Differentiate the expression (T dot t) + (N dot n) + (B dot b) with respect to s and show that you get zero. Conclude that (T dot t) + (N dot n) + (B dot b) is constant with respect to s.
• By calculating (T dot t) + (N dot n) + (B dot b) at the point p, show that the constant value is 3.
• Show that if (T dot t) + (N dot n) + (B dot b) = 3 for all s, then T = t, N = n, B = b for all s.
• Use the fact that T(s) = t(s) for all s, and that s is measured from the same initial point p to conclude that C_1 = C_2.

Challenge Problem 4
Let p and q be positive numbers for which (1/p) + (1/q) = 1. Show that for all x, y > 0 it is the case that (x^p)/p + (y^q)/q is greater than or equal to xy. Where does equality hold? (Hint: Set (x^p)/p + (y^q)/q = C and then maximize f(x,y) = xy subject to this restriction. Show that the maximum of f is less than or equal to C. Since this is true for any C, the inequality is established.)

Challenge Problem 5
Derive the familiar volume formulas for spheres and right circular cones by evaluating triple integrals in spherical coordinates.

Challenge Problem 6
Find the 4-dimensional `volume' enclosed by the four-sphere of radius r. (Which is the hypersurface defined by x^2 + y^2 + z^2 + w^2 = r^2.)

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