over the last week i have been trying to learn differential forms to try and tackle a problem, I believe I understand it, but I'm not quite sure, here is the problem:
Given a complex scalar field $\phi(x_1,x_2,x_3)$ defined on $\mathbb{R^3}$ which is well defined as $r\rightarrow \infty$, and $r=\sqrt{x_1^2 +x_2^2 +x_3^2}$
$\phi$ can be identified as $\phi : S^3 \rightarrow S^2$ $S^3 \equiv R^3 \cup \{\infty\}$ and $S^2 \equiv C^1 \cup \{\infty\}$ i.e the hopf map
the area two form is given by $F = \frac{1}{2\pi i}\frac{d\phi*\wedge d\phi}{(1+\phi*\phi)^2}$
is this expansion correct: $F =\frac{1}{2\pi i}(\frac{\partial \phi* }{\partial x_i}\frac{\partial \phi }{\partial x_j}-\frac{\partial \phi* }{\partial x_j}\frac{\partial \phi }{\partial x_i})(\frac{1}{(1+\phi*\phi)^2} ) dx_i \wedge dx_j $ where i $\neq$ j and x, j = 1, 2, 3
is $\phi$ considered as a 0-form?
$d\phi$ a 1-form in dual space $dx_i$?
and F an area in dual space (2-form)?
Thanks!