Directionality in Stokes Theorem for Volumes

K.J.Healey

I'm not sure if this post should go here or in the Calc setion, but I figure more knowledgable people browse this form. This question is relating to 'directionality' of doing closed loop integrals.
If you have some 2D wire structure, lets image it looks like a square wave, or a square well. Ok, square well, so we have a U shape, lets say with 4 points, ABCD where BC is the bottom of the well. Now if I were to sum the segments in the Y direction I would have AB + CD, and we can see that AB = DC = -CD
So we have AB - AB = 0.
Now rather then summing in the Y direction, were summing along a path. We define this path by the normals or tangentials (curl or div theorem respectively) of the segments. Since the N(ormal)ab = Ndc = -Ncd we have shown this works with the previous addition.
Now increase the rank so its summing surfaces to get a volume. Does the stoke's theorem incorporate a direction such that opposite sides of this cube-well now, when added together in say the Z direction are 0, but are , for the stokes, not added in directions but added with their normals taken into account? So like 2 opposite ends of the well would add rather than cancel?
I hope Im being clear. Its a difficult problem to visualize.

K.J.Healey

Nevermind. I realized my equation for the dissipation:
Int(T_i * u_(i,1) *ds) the ds is NOT n*dS, and the actual physics behind it lacks directionality. This creates another issue, but its more physics then mathematics. Is there a way to close/delete a thread?