Electromagnetics
nijirazdan i like your analogy.
here is the kicker.. your analogy makes it very clear that the energy associated with the curl vector (its moving, so must have energy) is entirely contained WITHIN the pipe.. the water can't leak out of the metal pipe after all..
mathematically speaking, this would be like integrating about the closed path.. where one looks at what is within the border of the integral.
here is the assumption with this analogy and math:
that there is NO energy outside the pipe wall, or if you give the benefit of the doubt, there is no "practically usable" energy outside the wall of the pipe.
what then is the significance of the Heaviside Component?
food for thought...