6760, Math 21a, Fall 2009
Exhibits page Math 21a 2009, Multivariable Calculus
Problem 12
Course head: Oliver Knill
Office: SciCtr 434

Problem 12

Extremizing the function
f(x,y,z) = x4 + y4 + z4
on the sphere
g(x,y,z) = x2 + y2 + z2 = 1
leads to many critical points: there are 26. For every vertex of the cube (v=8), for every face (f=6) and for every edge (e=12), there is a critical point. Now 8+6+12=26. Note that like for any polyhedron, one could have counted the edges with the formula
v-e+f  = 2 
which is the famous Euler polyhedra formula.

Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University