Let f be a continuous real-valued function on R3. Suppose that for every sphere S of radius 1, the integral of f(x, y, z) over the surface of S equals 0. Must f(x, y, z) = 0 for all points (x, y, z)?
Let f be a continuous real-valued function on R3. Suppose that for every sphere S of radius 1, the integral of f(x, y, z) over the surface of S equals 0. Must f(x, y, z) = 0 for all points (x, y, z)?
A single circle can change as the sphere moves even if the integral over the whole thing doesn't, though.
If it helps, the final answer is no, f doesn't have to be 0.
Hmm, a single circle isn't enough, though. It has to be the case for every single circle. I'll give it a rethink, though.
Not every circle independently, though, the parts as a whole.