icarus wrote: You really are beyond help (and a vicious, abusive jerk to boot).
And you are so niiice!
It is simple vector calculus that a field can be differentiable yet contain discontinuous regions, albeit infinitesimal. The forum medium does not lend itself to math symbols and I can't be bothered doing the 101 stuff for someone like you in ascii.
As usual you are missing the point. We are not talking here about
any scalar field but about the phase angle of a harmonic wave. Your problem is that you do not understand what a harmonic wave is and what the properties of its phase angle must be.
Although each point of the wave field has a phase angle which continuously changes with an angular frequency (omega), the way in which this phase angle changes with position is determined by the symmetry of the situation, and the boundary conditions which apply. What you are claiming is that one can model a harmonic wave by ignoring the symmetry and boundary conditions. This is again voodoo!
Just think of the line integral for the length of a circle in circular-polar coordinates ... at 2*pi the coordinates 'jump' or are multi-valued.
Yes let's do this. What you are saying is that the phase angle changes periodically from one point to the other
on and along the circular path so that there are crests and troughs along this circular path. What this means in physics is that you have a running wave moving along the circle.
This is not mathematically nor physically possible.
Let me try and explain it to you in simpler terms: If you go to a first year physics book, you will find that the phase angle only changes continuously with position when you have a coherent running wave. The change is then maximum in the directions perpendicular to the wavefronts (since one has that the undulations are moving in this direction) and zero parallel to the wave fronts because the wavefronts
cannot move parallel to themselves.
Now let us consider a supercurrent around a ring. You have circular symmetry. By taking a loop integral along the ring, as the "experts" do, they are modelling the current as if it is caused by wavefronts moving along the ring:
As already pointed out, this not possible.
What you first have to do is to solve the differential wave equation in terms of polar coordinates. Guess what!! The wave function becomes the product of an "angular wave function" and a "radial wave" function. This symmetry demands that, if there are wave fronts moving, they can only move out as circles: Thus there are no crests and troughs around such a circle. Thus there cannot be any crests and troughs along a circular path ever. The change in phase angle is totally determined by the angualr wave function.
And this is a hidden phase angle which only plays a role when you break the circular symmetry by, for example, applying a magnetic field at an angle to the symmetry axis. Otherwise it has
no effect whatsoever!
Anytime you introduce angles into your coordinates (almost inevitably) you get discontinuities, any physics done based on angular quantities cannot avoid these types of issues ..
This does not remove the simple fact that you
cannot calculate the change in phase angle as if it is driving a coherent harmonic wave around a ring. There are no crests and troughs along any such ring. Such a wave cannot model a current flowing around a ring as the "experts" on superconduction are claiming. I hope the penny has dropped.