Write $\mathcal A=aJ$ with the scalar one-form
$$
a=\langle dx,y\rangle=-\langle x,dy\rangle.
$$
The Lie algebra $\mathfrak o(2)$ is abelian, so $[\mathcal A\wedge\mathcal A]=0$. Moreover
$$
da((u_1,v_1),(u_2,v_2))
=\langle u_2,v_1\rangle-\langle u_1,v_2\rangle.
$$
Consequently
$$
\mathcal F((u_1,v_1),(u_2,v_2))
=\bigl(\langle u_2,v_1\rangle-\langle u_1,v_2\rangle\bigr)
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
$$
as required.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 3 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-115/3/e/solution