Condition for the $1$-form $f\omega$, being $f \in C(\Omega)$ and $\omega = \omega_1\text{d}x_1 + \omega_2\text{d}x_2$, to be closed.

Question

I am tryingto solve the following problem:

Let $\Omega$ be an open set of $\mathbb{R}^2$ and $\omega = \omega_1\text{d}x_1 + \omega_2\text{d}x_2 \in \text{F}_1(\Omega)$. Define $$ L = \omega_2 \frac{\partial}{\partial x_1}-\omega_1\frac{\partial}{\partial x_2}. $$ Show that
$\exists \ f \in C^{\infty}(\Omega), f > 0$ satisfying $ \text{d}(f\omega) = 0$ if and only if there exists exists $u \in C^{\infty}(\Omega)$ satifying $Lu = \frac{\partial\omega_2}{\partial x_1}-\frac{\partial\omega_1}{\partial x_2}$.

Now, I did the following:

Let $f$ have the desired properties. Then, since $f$ is a $0$-form: $$ \text{d}(f\omega) = \text{d}f \wedge \omega + (-1)^0f\wedge \text{d}\omega = \text{d}f \wedge \omega + f \text{d}\omega $$ $$ \therefore \text{d}(f\omega) = 0 \Longleftrightarrow \text{d}f\wedge\omega = -f\text{d}\omega $$ But: $$ \text{d}f = \frac{\partial f}{\partial x_1}\text{d}x_1+\frac{\partial f}{\partial x_2}\text{d}x_2 $$ $$ \therefore \text{d}f\wedge\omega = \left(\omega_2\frac{\partial f}{\partial x_1} - \omega_1\frac{\partial f}{\partial x_2}\right)\text{d}x_1\wedge\text{d}x_2 = Lf\text{d}x_1\wedge\text{d}x_2 $$ On the other side: $$ -f\text{d}\omega = -f\left(\frac{\partial \omega_2}{\partial x_1} -\frac{\partial \omega_1}{\partial x_2} \right)\text{d}x_1\wedge\text{d}x_2 $$ We then end up with $$ \text{d}(f\omega) = 0 \Longleftrightarrow Lf = -f \left(\frac{\partial \omega_2}{\partial x_1} -\frac{\partial \omega_1}{\partial x_2} \right) $$ which is almost what we wanted.

Any hints on what I may have missed or on where to go from here will be the most appreciated.

Thanks in advance.

Answer 1

With $f > 0$ and

$L = \omega_2 \dfrac{\partial}{\partial x_1} - \omega_1 \dfrac{\partial}{\partial x_2}, \tag 1$

it is easy to see that

$Lf = -f( \dfrac{\partial \omega_2 }{\partial x_1} - \dfrac{\partial \omega_1 }{\partial x_2}) \tag 2$

may be written

$-f^{-1}Lf = \dfrac{\partial \omega_2 }{\partial x_1} - \dfrac{\partial \omega_1 }{\partial x_2}; \tag 3$

also, with $f > 0$ we have

$-f^{-1}Lf = -L(\ln f), \tag 4$

since $L$ only involves first derivatives. Set

$u = -\ln f; \tag 5$

then

$Lu = L(-\ln f) = -f^{-1}Lf = \dfrac{\partial \omega_2 }{\partial x_1} - \dfrac{\partial \omega_1 }{\partial x_2} \tag 6$

as desired.

Answer 2

HINT: The fact that $f>0$ should be a hint. Try using $f=e^{-u}$ when $u$ satisfies the given equation.