Asymmetry in Stokes' theorem & Gauss' theorem

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Main Question or Discussion Point

Stokes theorem relates a closed line integral to surface integrals on any arbitrary surface bounded by the same curve. Gauss theorem relates a closed surface integral to the volume integral within a unique volume bounded by the same surface. What causes this asymmetry in these 2 theorems, in the language of calculus?
 

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  • #2
PeroK
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Stokes theorem relates a closed line integral to surface integrals on any arbitrary surface bounded by the same curve. Gauss theorem relates a closed surface integral to the volume integral within a unique volume bounded by the same surface. What causes this asymmetry in these 2 theorems, in the language of calculus?
A closed surface defines a unique volume. A closed curve defines a unique area in 2D, but not a unique 3D volume.
 
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A closed surface defines a unique volume. A closed curve defines a unique area in 2D, but not a unique 3D volume.
Right. Why is there such a difference? Mathematically or philosphically.
 
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PeroK
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Right. Why is there such a difference? Mathematically or philosphically.
There's no difference except the dimension. Why is a cube different from a square?
 
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A 2d closed surface can only enclose a unique 3d volume while a 1d close curve can enclose multiple 2d surfaces. Why’s that asymmetry?
 
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There is no asymmetry. A closed 2D surface can also enclose multiple 3D volumes if you allow "bridges", what you obviously did in the 2D case. The intersection point of a figure 8 can likewise be used to connect surfaces in 3D.
 
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There is no asymmetry. A closed 2D surface can also enclose multiple 3D volumes if you allow "bridges", what you obviously did in the 2D case. The intersection point of a figure 8 can likewise be used to connect surfaces in 3D.
Great. Could you please give an example of bridges for a spherical surface or anything simple?
 
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Just take two balls and connect them by a straight line. Then you have a closed surface. But even without, the two spheres alone are one closed surface. You confuse closed with connected.
 
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Just take two balls and connect them by a straight line. Then you have a closed surface. But even without, the two spheres alone are one closed surface. You confuse closed with connected.
Then in this example of yours, the 2 balls form 1 surface, then what is/are the enclosed 3d volume?
 
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Delta2
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A 2d closed surface can only enclose a unique 3d volume while a 1d close curve can enclose multiple 2d surfaces. Why’s that asymmetry?
I agree with you , this asymmetry exists indeed, and it must be something related to the topology of ##\mathbb{R^3}## and ##\mathbb{R^n}## in general. But as I was pretty weak in topology during my undergraduate studies, I cant seem to give a deeper explanation on why this asymmetry is present.
 
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I agree with you , this asymmetry exists indeed, and it must be something related to the topology of ##\mathbb{R^3}## and ##\mathbb{R^n}## in general. But as I was pretty weak in topology during my undergraduate studies, I cant seem to give a deeper explanation on why this asymmetry is present.
The reason I ask this is due to applications of Guass' and Stokes theorems.
 
  • #12
PeroK
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A 2d closed surface can only enclose a unique 3d volume while a 1d close curve can enclose multiple 2d surfaces. Why’s that asymmetry?
There is no asymmetry:

2D surface - unique 3D volume (in 3D space) - multiple 3D volumes (in 4D space)

1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)

It's the same thing one dimension higher.
 
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  • #13
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1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
[/QUOTE]
I doubt about this. What if the 1D curve isn't planar, then there's no 2D surface in 2D in your language.
 
  • #14
Delta2
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There is no asymmetry:

2D surface - unique 3D volume (in 3D space) - multiple 3D volumes (in 4D space)

1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)

It's the same thing one dimension higher.
Yes in ##R^4## there isnt such asymmetry but i believe there we would have another asymmetry present (something like a 3D volume can enclose only one 4D volume but a 3D surface can enclose many 4D volumes).
 
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PeroK
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1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
I doubt about this. What if the 1D curve isn't planar, then there's no 2D surface in 2D in your language.
[/QUOTE]

The difference between Gauss's law and Stokes's theorem is a mystery then! I don't know why they are not the same.
 
  • #16
PeroK
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Yes in ##R^4## there isnt such asymmetry but i believe there we would have another asymmetry present (something like a 3D volume can enclose only one 4D volume but a 3D surface can enclose many 4D volumes).
The original question is here:

https://www.physicsforums.com/threa...d-by-the-same-curve-in-stokes-theorem.989709/

And here:

https://www.physicsforums.com/threads/asymmetry-in-stokes-theorem-gauss-theorem.989780/

If you want to take a look. I may be misunderstanding what the OP is asking.
 
  • #17
PeroK
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1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
I doubt about this. What if the 1D curve isn't planar, then there's no 2D surface in 2D in your language.
[/QUOTE]

A non-planar 1D curve translates to a 2D surface embedded in 4D (not 3D). A circle translates to a sphere, but a close curve that extends into 3D translates to a closed 2D surface that extends into 4D.

You only get an asymmetry when you allow a 1D curve to be in 3D space, but restrict a 2D surface to 3D space.
 
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Then in this example of yours, the 2 balls form 1 surface, then what is/are the enclosed 3d volume?
The spheres are the 2D surfaces, the balls the 3D volume.
 
  • #19
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The spheres are the 2D surfaces, the balls the 3D volume.
Yes, so there's no more than 1 volume enclosed?
 
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Yes, so there's no more than 1 volume enclosed?
Why that? There are two spheres, hence they enclose two volumes / balls.
 
  • #21
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Why that? There are two spheres, hence they enclose two volumes / balls.
But when applying Gauss' theorem, one has to enclose both spheres. It's not a choice to enclose which of the 2.
 
  • #22
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But when applying Gauss' theorem, one has to enclose both spheres. It's not a choice to enclose which of the 2.
I think you still confuse closed and connected. But maybe I don't understand your point. A picture would be helpful.
 
  • #23
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I think you still confuse closed and connected. But maybe I don't understand your point. A picture would be helpful.
Could you post a pic of your idea of 1 2D surface enclosing more than 1 choice of 3D volume?
 
  • #24
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Could you post a pic of your idea of 1 2D surface enclosing more than 1 choice of 3D volume?
No, since I don't know what choice means in this context.
 
  • #25
pbuk
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Could you post a pic of your idea of 1 2D surface enclosing more than 1 choice of 3D volume?
Search "double cone".

You confusion arises because a simple closed curve bounds at most one* single continuous area whereas a closed curve that is not simple (i.e. crosses itself) may bound 2 or more disconnected areas.

Exactly the same is true for surfaces - a simple closed surface bounds at most one volume but a surface that crosses (i.e. intersects) itself may* bound 2 or more.

* note degenerate cases e.g. line on a Möbius strip, Klein bottle. Excercise - must a crossing curve enclose at least one area? What about a simple surface?
 

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