PROBLEM OF THE MONTH
October 2004
An n-dimensional ball of radius r is the set of points in Rn
that satisfy the equation
x12 + x22 + x32 + ... + xn2 &le r2
- Derive the following classical formulas :
- the volume of a 2-ball is Pi r2 .
(In two dimensions, "volume" is also called "area".)
- the volume of a 3-ball is (4 Pi / 3) r3.
- Show that the volume of the n-dimensional ball is
V = cn rn
where cn is a constant that depends only on the dimension, n.
- Show that the constants cn satisfy the recurrence relation
cn = (2 Pi / n) cn-2
This month's prize will be award to the best explained, correct solution.
Submit your solution to the Mathematics Undergraduate Office (Math P-142)
or electronically to Prof. Kudzin at
problem@math.sunysb.edu
by the due date. Acceptable electronic formats are: PDF, Postscript, DVI,
(La)TeX, or just plain text. Please include your name and phone number,
or preferably your email address.
Closing date: November 1st at 12 pm.