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 knowledgeable people browse this form. This question is relating to 'directionality' of doing closed loop integrals.
If you have some 2D wire structure, let's image it looks like a square wave, or a square well. Ok, square well, so we have a U shape, let's 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 I am 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?

 

[M98Ranger]

The directionality in Stokes Theorem for volumes is an important aspect to consider when using this theorem to calculate surface integrals. In the example given, the directionality is shown through the use of normals and tangents to define the path along which the integral is taken. This is necessary because in three-dimensional space, there are multiple ways to approach a surface and the direction in which the integral is taken can affect the final result.

In the case of the square well, if the integral is taken in the Y direction, as in the example, the result is 0 because the opposite sides of the well cancel each other out. However, if the integral is taken in the Z direction, the opposite sides would add together and the result would not be 0. This is because the direction of the integral changes and the opposite sides are no longer being added in opposite directions.

In general, Stokes Theorem incorporates directionality by taking into account the orientation of the surface being integrated over. This is done through the use of the normal vector to the surface, which represents the direction in which the surface is oriented. By considering the orientation of the surface, the theorem is able to properly account for the direction in which the integral is taken and give an accurate result.

It is important to keep in mind that the directionality in Stokes Theorem is not limited to just the direction in which the integral is taken, but also the direction of the normal vector to the surface being integrated over. This is why it is crucial to carefully define the path of integration and the orientation of the surface when using this theorem to calculate surface integrals.