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Forms on the circle 
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#1
Dec209, 01:30 AM

P: 333

I read a problem a while ago which was to find a differential form on the circle which is not the differential of any function. Being a hapless physicist, this puzzled me for a while. I've found an answer in Spivak's Calculus on Manifolds, but I need a little help in following his reasoning.
He argues that the form [tex]d\theta[/tex] is such a form, and shows that if it is the differential of a function [tex]f[/tex] then [tex]f = \theta + constant[/tex]. I am OK up to this point, but I fail to see how such an [tex]f[/tex] can't exist, like he argues. 


#2
Dec209, 02:44 AM

Sci Advisor
P: 1,588

The function
[tex]f(\theta) = \theta[/tex] is not singlevalued on the circle. Note that every closed form can be written as d of something locally; that is, over some finite region of the manifold. But only exact forms can be written as d of something globally. Here is an example on the circle of a form that is closed but not exact. Another example that you might look up is the Dirac monopole. It can be written as [itex]\vec B = \vec \nabla \times \vec A[/itex] over some region of space, but not globally. There is always a topological defect (the Dirac string) on which [itex]\vec A[/itex] is not defined. 


#3
Jan610, 07:31 AM

P: 707

The angle function ,theta, is discontinuous and so does not have a derivative everywhere. Where it is continuous though, its differential equals dtheta. 


#4
Jan610, 08:20 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348

Forms on the circle



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