On 2013-03-26, at 1:36 PM, Andy Latto wrote:
Well, if we're working in 3 dimensions, it's a surface integral, not a line integral.
(sorry to repeat, but:) No, a surface has a boundary (which may be empty). The integral in question is a line integral around the boundary of the surface. That's why the *surface* integral on the other side of the equation is 0 when the surface is closed, i.e., has empty boundary.
You mean
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.
But the unit sphere is only one of the two components of the boundary of our region.
Who mentioned a 3D region? I didn't. Just a surface with empty boundary. --Dan