I struggle to understand the following question. I expect I'm simply being dense about something.
Let $F$ be a vector field on $U$, open in $\mathbb R^3,$ $F = \sum_1^3 f_j (x) \frac{\partial}{\partial x_j}$. Consider the $1$-form $\varphi=\sum_1^3 f_j(x) dx_j.$ Show that $d \varphi$ and $\operatorname{curl} F$ are related in the following way:
$$\operatorname{curl} F = \sum_1^3 g_j(x) \frac{\partial}{\partial x_j}$$
$$d\varphi =g_1(x) dx_2 \wedge dx_3 + g_2(x) dx_3 \wedge dx_1 + g_3(x)dx_1 \wedge dx_2$$
What are the $g_i$?
For reference, this is Exercise 5 in section 13 of Chapter 1.
They are simply what you get when you compute the curl (and the differential): $g_1=\partial f_3/\partial x_2-\partial f_2/\partial x_3$, and so on.