How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ?
$$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$
Where $f, g\in C^1$ are homogeneous functions with the same degree $\lambda>0$, in sense that $f(tz)=t^{\lambda}f(z)$ and the same for $g$. And $A=\{(x,y)|xf(x,y)+yg(x,y)\neq 0\}.$ Sorry for my English. Any help is welcome.
Note that, if
$$A(x,y)=\frac{f(x,y)}{xf(x,y)+yg(x,y)}$$ and $$B(x,y)=\frac{f(x,y)}{xf(x,y)+yg(x,y)},$$
you can derive and find that
$$A_y=\frac{yf_yg-fg-yfg_y}{(xf+yg)^2}$$
and
$$B_x=\frac{xfg_x-fg-xf_xg}{(xf+yg)^2}.$$
Theorem: If $\omega=Adx+Bdy$ a 1-form is $C^1$ such that $A_y=B_x$, then $\omega$ is closed.
Hint: Use that $f$ and $g$ are homogeneous functions to conclude that.