Any surface bounded by the same curve in Stokes' theorem

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In Stokes' theorem, the closed line integral of f=the surface integral of curl f on ANY surface bounded by the same curve. But in Gauss' theorem, the surface integral of f on a surface=the volume integral of div f on a unique volume bounded by the surface. A surface can only enclose 1 volume whereas a curve can enclose many surfaces. So why is the asymmetry?
 

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  • #2
wrobel
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it is because in ##\mathbb{R}^3## if a 2D surface bounds some volume then this volume is unique. It is just a feature of ##\mathbb{R}^3##. The general Stokes formula ##\int_{\partial M}\omega=\int_Md\omega## is symmetric. And the manifold ##M## is not obliged to be embedded anywhere
 
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  • #3
WWGD
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it is because in ##\mathbb{R}^3## if a 2D surface bounds some volume then this volume is unique. It is just a feature of ##\mathbb{R}^3##. The general Stokes formula ##\int_{\partial M}\omega=\int_Md\omega## is symmetric. And the manifold ##M## is not obliged to be embedded anywhere
I'm curious on this property of ##\mathbb R ^3##. Do you mean that its homology is trivial?
 
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it is because in ##\mathbb{R}^3## if a 2D surface bounds some volume then this volume is unique. It is just a feature of ##\mathbb{R}^3##. The general Stokes formula ##\int_{\partial M}\omega=\int_Md\omega## is symmetric. And the manifold ##M## is not obliged to be embedded anywhere
Could you please explain why this asymmetry (1D-2D and 2D-3D) is in the language of calculus or on a similar level?
 
  • #5
wrobel
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Could you please explain why this asymmetry (1D-2D and 2D-3D) is in the language of calculus or on a similar level?
If you live in 2d world there only one 2d-volume is contained inside a loop of a closed curve. But if you exit in 3d then you see that there are a lot of two dimensional films pulled up on this curve. The same if you come out from ##\mathbb{R}^3## to ##\mathbb{R}^4##. The general Stokes formula does not care in which ##\mathbb{R}^m## you live and which metric you use to consider vectors instead of differential forms. Have a patience it will be explained in more advanced courses later
 
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If you live in 2d world there only one 2d-volume is contained inside a loop of a closed curve. But if you exit in 3d then you see that there are a lot of two dimensional films pulled up on this curve. The same if you come out from ##\mathbb{R}^3## to ##\mathbb{R}^4##. The general Stokes formula does not care in which ##\mathbb{R}^m## you live and which metric you use to consider vectors instead of differential forms. Have a patience it will be explained in more advanced courses later
I can think of only 1 possibility where a 2d surface can enclose multiple 3d volumes: the 3d volume inside or that outside (cavity). Does that count as an analogue of 1d curve enclosing multiple 2d surfaces?
 
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The question has been answered.
 

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