- #1
Poopsilon
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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 [tex]\eta = \frac{xdy-ydx}{x^2+y^2}[/tex] on the set [tex]\mathbb{R}^2-{0}[/tex] and then parametrizes the circle [tex]\gamma(t)=(rcos(t),rsin(t))[/tex] for fixed r>0 and 0≤t≤2pi.
Now[tex]d\eta=0[/tex]yet direct computation shows that [tex]\int_{\gamma}\eta=2\pi[/tex].
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 [tex]\mathbb{R}^3-{0}[/tex] 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 [tex]\int_{\gamma}\eta=0[/tex] thus contradicting the integral given above.
Also what is up with this C'' requirement? I can't figure out why its important.
Now[tex]d\eta=0[/tex]yet direct computation shows that [tex]\int_{\gamma}\eta=2\pi[/tex].
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 [tex]\mathbb{R}^3-{0}[/tex] 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 [tex]\int_{\gamma}\eta=0[/tex] thus contradicting the integral given above.
Also what is up with this C'' requirement? I can't figure out why its important.
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