I'm trying to read Lectures on quasiconformal mappings by Ahlfors, and I have trouble understanding some notation. I have a very basic understanding of differential geometry, so bear with me here.
On page 6 he takes a function $f(z) = u + iv$, $z = x + iy$, and writes
$du = u_x dx + u_y dy$
$dv = v_x dx + v_y dy$
and then says "in classical notation one writes"
$du^2 + dv^2 = (u_x^2 + v_x^2) dx^2 + (2u_x u_y + 2 v_x v_y) dx dy + (u_y^2 + v_y^2) dy^2$
I would like to thoroughly go through this as an exercise in differential geometry.
My understanding of the first equations is this: we have the tangent spaces at a specific point $z$ and $f(z)$, both of which are $\mathbb{R}^2$. We have the differential forms $dx, dy$ on the tangent space of the domain of $f$, which act by projecting a vector onto first/second coordinate, and these linear transformations between $\mathbb{R}^2$ and $\mathbb{R}$ obviously also form a linear space. The function $f$ induces a linear transformation between these spaces, and it can be shown that it's given by these first two equations.
Now I'm a bit lost what he means here by $du^2$, because I don't think it's literally taking the square of $u$? And what object is $dx dy $, is that a wedge product? Basically the entire meaning of the last equation escapes me, thanks in advance.