Consider the following differential forms in $\mathbb{R}^{3}$ :

$$\alpha=x d x+y d y+z d z, \quad \beta=z d x+x d y+y d z, \quad \gamma=x y d z$$

(i) Is $\alpha$ closed or exact? Is $\gamma$ closed or exact?

(ii) Calculate $\alpha \wedge \beta$ and $(\alpha+\gamma) \wedge(\alpha+\gamma)$.