Closed and Exact Forms on 2-Torus: Solving for Global Definitions and Exactness

[Breo]

Homework Statement

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?

Homework 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: $$ \int_{C_a} \omega = 0$$

The Attempt at a Solution

[/B]
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.

 

[stevendaryl]

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:
$$
-B d\alpha_1 \wedge \alpha_2 = \frac{\partial \phi (\alpha_1, \alpha_2)}{\partial \alpha_1}d\alpha_1 \wedge \alpha_2 $$

What happened to the terms involving $C$ and $D$?

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

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

c) No idea. I get lost when read the de Rham cohomology theory, etc.

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