This question is directed to everyone whose initials are not E.S. or V.E. Assume any vector fields or surface mentioned is at least C^2. Or even real analytic. The symbol == means "equals for all values of the domain". Under Divergenceless Field[1] at MathWorld, the first sentence asserts that (*) div(V) == 0 implies that there exists a vector field W such that V == curl(W). But their entry for Curl Theorem[2] states that the flux of a vector field of the form curl(W) through a surface S is equal to the line integral of W around its boundary bd(S). In particular, this implies: (**) The flux of curl(W) through a closed surface must be 0, for any vector field W. Now consider the vector field V given by V(x,y,z) = (x,y,z) / (x^2 + y^2 + z^2)^(3/2), (x,y,z) unequal to (0,0,0). It's easy to check that div(V) == 0. By (*) there exists a W such that V == curl(W). But it's also easy to check that the flux of V through the unit sphere x^2 + y^2 + z^2 = 1 is 4pi. This contradicts (**). PUZZLE: Explain how this paradox is possible. --Dan _____________________________________________________________ [1] < http://mathworld.wolfram.com/DivergencelessField.html > [2] < http://mathworld.wolfram.com/CurlTheorem.html >