Asymmetry in Stokes' theorem & Gauss' theorem

In summary, Stokes theorem relates a closed line integral to surface integrals on any arbitrary surface bounded by the same curve, while Gauss theorem relates a closed surface integral to the volume integral within a unique volume bounded by the same surface. The difference in dimensionality between curves and surfaces causes the apparent asymmetry in these two theorems. However, this is not a true asymmetry, as the same pattern is seen one dimension higher. Additionally, the concept of a closed surface can be extended to multiple volumes if "bridges" are allowed, similar to how a figure 8 can connect multiple surfaces in 3D. The reason for this difference may be related to the topology of different dimensional spaces, but this is not fully understood.
  • #1
feynman1
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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?
 
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  • #2
feynman1 said:
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.
 
  • #3
PeroK said:
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.
 
  • #4
feynman1 said:
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?
 
  • #5
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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  • #6
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.
 
  • #7
fresh_42 said:
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?
 
  • #8
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.
 
  • #9
fresh_42 said:
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?
 
  • #10
feynman1 said:
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 can't seem to give a deeper explanation on why this asymmetry is present.
 
  • #11
Delta2 said:
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 can't 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
feynman1 said:
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
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
PeroK said:
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 isn't 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).
 
  • #15
feynman1 said:
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
Delta2 said:
Yes in ##R^4## there isn't 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
feynman1 said:
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.
 
  • #18
feynman1 said:
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
fresh_42 said:
The spheres are the 2D surfaces, the balls the 3D volume.
Yes, so there's no more than 1 volume enclosed?
 
  • #20
feynman1 said:
Yes, so there's no more than 1 volume enclosed?
Why that? There are two spheres, hence they enclose two volumes / balls.
 
  • #21
fresh_42 said:
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
feynman1 said:
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
fresh_42 said:
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
feynman1 said:
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
feynman1 said:
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?
 
  • #26
pbuk said:
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?
Thank you. How would you explain 1 surface enclosing only 1 volume in applying Gauss' theorem? How will it apply to a double cone?
 
  • #27
feynman1 said:
Thank you. How would you explain 1 surface enclosing only 1 volume in applying Gauss' theorem?
I wouldn't. The divergence theorem (which is how we now refer to the vector calculus generalisation of Gauss's law in electrostatics), is not restricted to simple closed surfaces.
feynman1 said:
How will it apply to a double cone?
Well it would have to be a double truncated cone to be closed. Why do you think there is a difficulty in applying Gauss's law here?
 
  • #28
pbuk said:
I wouldn't. The divergence theorem (which is how we now refer to the vector calculus generalisation of Gauss's law in electrostatics), is not restricted to simple closed surfaces.

Well it would have to be a double truncated cone to be closed. Why do you think there is a difficulty in applying Gauss's law here?
My difficulty is, for Stokes' theorem 1 curve can enclose multiple surfaces but in Gauss' 1 surface encloses 1 volume
 
  • #29
feynman1 said:
My difficulty is, for Stokes' theorem 1 curve can enclose multiple surfaces but in Gauss' 1 surface encloses 1 volume
You keep stating this but you are not responding to a number of different people pointing out that a single "volume" in what you call Gauss's law is not necessarily continuous.
 
  • #30
pbuk said:
You keep stating this but you are not responding to a number of different people pointing out that a single "volume" in what you call Gauss's law is not necessarily continuous.
Also, this is the third post on the subject. It was already answered in the threads mentioned in post #12 above. Here, for example, is a good answer to the question:

https://www.physicsforums.com/threa...d-by-the-same-curve-in-stokes-theorem.989709/
 
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  • #31
PeroK said:
Also, this is the third post on the subject. It was already answered in the threads mentioned in post #12 above. Here, for example, is a good answer to the question:

https://www.physicsforums.com/threa...d-by-the-same-curve-in-stokes-theorem.989709/
@feynman1 You apparently got your answer. If you still have problems, then we may have communication problems, which is not unlikely since we are restricted to verbal only communication.

My suggestion is: Quote the original theorem you have a question to. Famous theorems are sometimes differently phrased. It looks as if you have trouble to understand the conditions of Gauß law. So the more precise you ask, the better the answers will be.
 
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Related to Asymmetry in Stokes' theorem & Gauss' theorem

1. What is asymmetry in Stokes' theorem and Gauss' theorem?

Asymmetry in Stokes' theorem and Gauss' theorem refers to the fact that these two theorems involve different types of integrals and have different orientations. In Stokes' theorem, the integral is a line integral, while in Gauss' theorem, the integral is a surface integral. Additionally, the orientation of the surface or curve in the integral also differs between the two theorems.

2. How do Stokes' theorem and Gauss' theorem relate to each other?

Stokes' theorem and Gauss' theorem are both fundamental theorems in vector calculus that relate the behavior of a vector field in three-dimensional space to the behavior of the field on the boundary of a region in three-dimensional space. While they are different theorems, they are both based on the concept of flux, which measures the flow of a vector field through a surface or curve.

3. What is the significance of asymmetry in Stokes' theorem and Gauss' theorem?

The asymmetry in Stokes' theorem and Gauss' theorem is significant because it allows for theorems to be applied to different types of problems. Stokes' theorem is typically used to evaluate line integrals, while Gauss' theorem is used to evaluate surface integrals. This allows for a wider range of applications and makes these theorems powerful tools in mathematics and physics.

4. How are Stokes' theorem and Gauss' theorem used in real-world applications?

Stokes' theorem and Gauss' theorem are used in a variety of real-world applications, particularly in physics and engineering. For example, they are used to calculate the flow of fluids through pipes, the electric field around a charged object, and the magnetic field around a current-carrying wire. They are also used in fluid mechanics, electromagnetism, and other fields of physics to solve problems involving vector fields.

5. Are there any limitations to the use of Stokes' theorem and Gauss' theorem?

While Stokes' theorem and Gauss' theorem are powerful tools, they do have some limitations. They can only be applied to continuous vector fields and regions with smooth boundaries. In addition, they are limited to three-dimensional space and cannot be applied to higher dimensions. It is also important to note that these theorems are based on certain assumptions and may not be applicable in all situations.

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