# Asymmetry in Stokes' theorem & Gauss' theorem

• I

## 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?

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.

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.

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?

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?

• Delta2
fresh_42
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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.

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?

fresh_42
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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.

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?

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.

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.

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.

• 1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
[/QUOTE]

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).

PeroK
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1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
[/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.

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:

If you want to take a look. I may be misunderstanding what the OP is asking.

PeroK
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1D curve - unique 2D surface (in 2D) - multiple 2D surfaces (in 3D)
[/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.

fresh_42
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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.

The spheres are the 2D surfaces, the balls the 3D volume.
Yes, so there's no more than 1 volume enclosed?

fresh_42
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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.

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.

fresh_42
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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.

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?

fresh_42
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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.

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?