Differential forms and Stokes' theorem

Evaluate the 2-form $dx \wedge dy$ on the pair of vectors $u = (1, 2)$, $v = (3, 4)$, in that order.

Which of these forms on $\mathbb{R}^3$ are equal to $0$? (Choose all that apply.)

Compute $d(x^2 y\, dz)$ on $\mathbb{R}^3$.

Let $R$ be the rectangle $[0, 2] \times [0, 3]$ with its boundary oriented anticlockwise. Use Stokes' theorem to compute $\oint_{\partial R} (x\,dy - y\,dx)$.

Green's theorem $\iint_D (Q_x - P_y)\,dx\,dy = \oint_{\partial D} P\,dx + Q\,dy$ is the general Stokes theorem $\int_M d\omega = \int_{\partial M}\omega$ with which choice of $M$ and $\omega$?

Away from the $y$-axis, $d\theta = d\big(\arctan(y/x)\big)$. Why does this not show that $d\theta$ is exact on $\mathbb{R}^2 \setminus \{0\}$?

Which Lean statement correctly says that $d\theta$ is not exact on the punctured plane? (dTheta p is the linear functional $d\theta$ at the point p.)

A 1-form on $\mathbb{R}^3$ is a triple $(a_1, a_2, a_3)$ meaning $a_1\,dx + a_2\,dy + a_3\,dz$, and a 2-form is a triple $(c_1, c_2, c_3)$ meaning $c_1\,dy\wedge dz + c_2\,dz \wedge dx + c_3\,dx \wedge dy$. Write wedge(a, b) returning the 2-form $a \wedge b$ as such a triple. What familiar operation do you get?

On which of these open sets is every closed 1-form exact? (Choose all that apply.)