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Exact and closed forms

  1. Nov 25, 2014 #1
    1. The problem statement, all variables and given/known data

    Now consider a 2-torus ## S_1 × S_1## and a coordinate patch with coordinates ## (\alpha_1, \alpha_2)## such that ## 0 < \alpha_i < 2 \pi##. Let us introduce in this patch a 1-form of the type:
    $$\omega = (A + B\alpha_2 + C sin(\alpha_2 ) + D cos(2\alpha_1 + \alpha_2 ))d\alpha_1 + \phi(\alpha_1 , \alpha_2 )d\alpha_2$$

    a) Try to determine the possible values of the function ##\phi(\alpha_1 , \alpha_2 )## so
    that the form ##\omega## is closed.

    b) For what values of A, B, C and D is the closed form globally
    defined?

    c) For what values of A, B, C, D and ##\phi## is the form exact?



    2. Relevant equations

    A form is closed when satisfies: ##d\omega=0##

    The necessary and sufficient condition for a closed r-form ##\omega## to be exact is that for ##b_r## independent cycles in ##C_a \in H_r## the periods vanish: $$ \int_{C_a} \omega = 0$$


    3. The attempt at a solution


    a) Using the equation of the exterior derivative and knowing that ##d\omega = 0## I wrote:

    $$ 0 + (\frac{\partial B\alpha_2}{\partial \alpha_1}d\alpha_1 + \frac{\partial B\alpha_2}{\partial \alpha_2}d\alpha_2)\wedge \alpha_1 + .... = \Bigg( \frac{\partial \phi (\alpha_1, \alpha_2)}{\partial \alpha_1}d\alpha_1 + \frac{\partial \phi (\alpha_1, \alpha_2)}{\partial \alpha_2}d\alpha_2 \Bigg) \wedge d\alpha_2$$

    Using the Poincarè lemma ##d^2=0## and the relation ## dx^i \wedge dx^j = -dx^j \wedge dx^i ## I finally obtained:
    $$
    -B d\alpha_1 \wedge \alpha_2 = \frac{\partial \phi (\alpha_1, \alpha_2)}{\partial \alpha_1}d\alpha_1 \wedge \alpha_2 $$

    Not sure how to resolve this, maybe: ## \phi (\alpha_1, \alpha_2) = -2\pi B ## ?

    b) Do not know... it is when the coord charts are defined for all values of a parameter? Sorry I do not know.

    c) No idea. I get lost when read the de Rham cohomology theory, etc.
     
  2. jcsd
  3. Nov 30, 2014 #2

    stevendaryl

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    Staff Emeritus
    Science Advisor

    What happened to the terms involving [itex]C[/itex] and [itex]D[/itex]?

    I'm not 100% positive what the question is asking for, but if [itex]\alpha_1[/itex] and [itex]\alpha_2[/itex] are angles, then that means that [itex]\alpha_1 = 0[/itex] is the same angle as [itex]\alpha_1 = 2 \pi[/itex], and similarly for [itex]\alpha_2[/itex]. That means that a globally defined function should be periodic in [itex]\alpha_1[/itex] and [itex]\alpha_2[/itex].

    A 1-form [itex]\omega[/itex] is exact if there is a function [itex]f(\alpha_1, \alpha_2)[/itex] such that [itex]\omega = d f[/itex]. So you're just being asked when it is possible to find such an [itex]f[/itex].
     
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