I'm having trouble proving the formula for a change of variables under a double integral, described by the transformation of $u$ and $v$: $$x = g(u,v)$$ $$y = h(u,v)$$
The statement to be proven:
$$\int\int_{R} dx dy$$ $$= \int\int_{S} (g_u h_v- g_v h_u) du dv $$
Subscripts denote partial derivatives. I attempted to prove it by first writing $dx$ and $dy$:
$$dx = g_udu +g_v dv$$ $$dy = h_udu + h_vdv$$
Therefore,
$$\int\int_{R} dx dy$$
$$ = \int\int_{S} (g_udu +g_v dv)(h_udu + h_vdv) $$
$$ \int\int_{S} [g_uh_udu^2 + (g_u h_v + g_v h_u)du dv + g_vh_v dv^2] $$
For reference, here is the correct integrand: $$g_u h_v- g_v h_u$$
My answer looks very different, so what am I doing wrong?