Why does Stokes' theorem give different results for γ in ℝ² vs ℝ³?

[Poopsilon]

So I'm going over Rudin's chapter on differential forms in his Principles of Mathematical Analysis and I'm looking at Example 10.36 which gives the 1 form $$\eta = \frac{xdy-ydx}{x^2+y^2}$$ on the set $$\mathbb{R}^2-{0}$$ and then parametrizes the circle $$\gamma(t)=(rcos(t),rsin(t))$$ for fixed r>0 and 0≤t≤2pi.

Now$$d\eta=0$$yet direct computation shows that $$\int_{\gamma}\eta=2\pi$$.

Thus by Stokes' Theorem we can then conclude that gamma is not the boundary of any 2-chain of class C'' in the punctured plane.

Now I understand that because the origin is not included we can't just use the disk of radius r as our 2-surface with boundary equal to gamma (I think this is related to the Cauchy Residue Theorem) but what if we extended ourselves to $$\mathbb{R}^3-{0}$$ than we could parametrize some sort of cone-like 2-surface with boundary equal to gamma which we could probably make C'' which would then by Stokes' Theorem force the integral to be $$\int_{\gamma}\eta=0$$ thus contradicting the integral given above.

Also what is up with this C'' requirement? I can't figure out why its important.

 

[quasar987]

The formula for eta as it stands can be interpreted as a 1-form on R² - 0 or R³ - "z axis". If you want a 1-form on R³ - 0, you need to specify in addition what eta is to be on "z-axis" - 0. But you're going to have trouble doing that in a continuous fashion!

 

[Poopsilon]

Can't I just leave it is as so it doesn't even involve the z axis just sits inside the xy-plane?

 

[quasar987]

We if you're going to talk about a 1-form on R³ - 0, you better tell what its value is on each point of R³ - 0. Otherwise, that's not a 1-form on R³ - 0.

It's like if you say "consider the function f: R-->R defined by f(0)=0". That's nonsense: the formula f(0)=0 only defines a map f:{0}-->R.

 

[Poopsilon]

Ah Sorry you are completely right the interpretation I will have to have is R³ - 'z-axis' which would disallow any type of cone structure with boundary gamma. Could you tell me why the C'' condition is important for my surface, I mean C' I understand because you need to be able to take the Jacobian upon changing variables, but why C''?

 

[quasar987]

I don't know about the C² thing.

 

[lavinia]

I think that technically you would need to show that eta has no extension to a closed 1 form in R^3 - 0.

Try pulling such a form back to the unit sphere via the inclusion map and then reducing the problem to the planar case.