Any surface bounded by the same curve in Stokes' theorem

[feynman1]

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?

 

[wrobel]

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

 

[WWGD]

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?

 

[feynman1]

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?

 

[wrobel]

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

 

[feynman1]

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?

 

[fresh_42]

The question has been answered.