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)?
Dont trust me I havent taken math for a few years. Any non-zero f with div f = 0 should satisfy the condition, by divergence theorem. So no.
I think this wouldn't work since it looks like f is from R3 -> R. Divergence would only apply if the range of f was also in R3 afaik. I havent been able to think up a counterexample personally.